Non-Noether symmetries in Hamiltonian Dynamical Systems

George Chavchanidze
Department of Theoretical Physics,A. Razmadze Institute of Mathematics,1 Aleksidze Street, Tbilisi 0193, Georgia
abstract. We discuss geometric properties of non-Noether symmetries andtheir possible applications in integrable Hamiltonian systems.Correspondence between non-Noether symmetries and conservation lawsis revisited. It is shown that in regular Hamiltonian systemssuch symmetries canonically lead to Lax pairs on the algebraof linear operators on cotangent bundle over the phase space.Relationship between non-Noether symmetries and other widespread geometricmethods of generating conservation laws such as bi-Hamiltonian formalism,bidifferential calculi and Frölicher-Nijenhuis geometry is considered.It is proved that the integrals of motion associated with acontinuous non-Noether symmetry are in involution whenever thegenerator of the symmetry satisfies a certain Yang-Baxter type equation.Action of one-parameter group of symmetry on algebra of integrals of motionis studied and involutivity of group orbits is discussed.Hidden non-Noether symmetries of Toda chain, Korteweg-de Vries equation,Benney system, nonlinear water wave equations and Broer-Kaup systemare revealed and discussed.
keywords. Non-Noether symmetry; Conservation law; bi-Hamiltonian system; Bidifferential calculus; Lax pair; Frölicher-Nijenhuisoperator; Korteweg-de Vries equation; Broer-Kaup system; Benney system; Toda chain
msc. 70H33; 70H06; 58J70; 53Z05; 35A30

Introduction

Symmetries play essential role in dynamical systems, because they usually simplifyanalysis of evolution equations and often provide quite elegant solution of problems that otherwise wouldbe difficult to handle. In Lagrangian and Hamiltonian dynamical systems special role is playedby Noether symmetries — an important class of symmetries that leave action invariantand have some exceptional features. In particular, Noether symmetries deservedspecial attention due to celebrated Noether's theorem, that established correspondencebetween symmetries, that leave action functional invariant, and conservation lawsof Euler-Lagrange equations. This correspondence can be extended to Hamiltoniansystems where it becomes more tight and evident then in Lagrangian case and gives riseto Lie algebra homomorphism between Lie algebra of Noether symmetries and algebra ofconservation laws (that form Lie algebra under Poisson bracket).
Role of symmetries that are not of Noether type has been suppressed for quite a long time.However, after some publications of Hojman, Harleston, Lutzky and others(see [16], [36], [39],[40], [49]-[57])it became clear that non-Noether symmetries also can play important role inLagrangian and Hamiltonian dynamics. In particular, according to Lutzky[51], in Lagrangian dynamics there is definite correspondence between non-Noether symmetries andconservation laws. Moreover, each generator of non-Noether symmetrymay produce whole family of conservation laws (maximal number of conservation laws that canbe associated with non-Noether symmetry via Lutzky's theorem is equal to the dimension ofconfiguration space of Lagrangian system). This fact makes non-Noether symmetries especiallyvaluable in infinite dimensional dynamical systems, where potentially one can recoverinfinite sequence of conservation laws knowing single generator of non-Noether symmetry.
Existence of correspondence between non-Noether symmetries and conserved quantitiesraised many questions concerning relationship among this type of symmetries andother geometric structures emerging in theory of integrable models.In particular one could notice suspicious similarity between the method of constructingconservation laws from generator of non-Noether symmetry andthe way conserved quantities are produced in either Lax theory, bi-Hamiltonian formalism,bicomplex approach or Lenard scheme.It also raised natural question whether set of conservation laws associated with non-Noethersymmetry is involutive or not, and since it appeared that in general it may not be involutive,there emerged the need of involutivity criteria, similar to Yang-Baxter equation used in Lax theoryor compatibility condition in bi-Hamiltonian formalism and bicomplex approach.It was also unclear how to construct conservation laws in case of infinite dimensionaldynamical systems where volume forms used in Lutzky's construction are no longer well defined.Some of these questions were addressed in papers [11]-[14],while in the present review we would like to summarize all these issues and to provide someexamples of integrable models that possess non-Noether symmetries.
Review is organized as follows. In first section we briefly recall some aspects of geometricformulation of Hamiltonian dynamics. Further, in second section, correspondencebetween non-Noether symmetries and integrals of motion in regular Hamiltonian systems isdiscussed. Lutzky's theorem is reformulated in terms of bivector fieldsand alternative derivation of conserved quantities suitable for computations in infinitedimensional Hamiltonian dynamical systems is suggested. Non-Noether symmetries oftwo and three particle Toda chains are used to illustrate general theory.In the subsequent section geometric formulation of Hojman's theorem [36]is revisited and some examples are provided. Section 4 reveals correspondence betweennon-Noether symmetries and Lax pairs. It is shown that non-Noether symmetry canonicallygives rise to a Lax pair of certain type. Lax pair is explicitly constructed in termsof Poisson bivector field and generator of symmetry. Examples of Toda chains are discussed.Next section deals with integrability issues. An analogue of Yang-Baxter equationthat, being satisfied by generator of symmetry, ensures involutivity of setof conservation laws produced by this symmetry, is introduced.Relationship between non-Noether symmetries and bi-Hamiltonian systemsis considered in section 6. It is proved that under certain conditions,non-Noether symmetry endows phase space of regular Hamiltonian system withbi-Hamiltonian structure. We also discuss conditions under which non-Noethersymmetry can be "recovered" from bi-Hamiltonian structure.Theory is illustrated by example of Toda chains. Next section is devoted tobicomplexes and their relationship with non-Noether symmetries. Special kindof deformation of De Rham complex induced by symmetry is constructed in terms ofPoisson bivector field and generator of symmetry.Samples of two and three particle Toda chain are discussed.Section 8 deals with Frölicher-Nijenhuis recursion operators.It is shown that under certain condition non-Noether symmetrygives rise to invariant Frölicher-Nijenhuis operator on tangentbundle over phase space.The last section of theoretical part contains some remarks on action of one-parametergroup of symmetry on algebra of integrals of motion. Special attention is devoted toinvolutivity of group orbits.
Subsequent sections of present review provide examples of integrable modelsthat possess interesting non-Noether symmetries. In particular section 10 revealsnon-Noether symmetry of n-particle Toda chain. Bi-Hamiltonian structure,conservation laws, bicomplex, Lax pair and Frölicher-Nijenhuis recursionoperator of Toda hierarchy are constructed using this symmetry.Further we focus on infinite dimensional integrable Hamiltonian systems emergingin mathematical physics. In section 11 case of Korteweg-de Vriesequation is discussed. Symmetry of this equation is identified and used in constructionof infinite sequence of conservation laws and bi-Hamiltonian structure ofKdV hierarchy. Next sectionis devoted to non-Noether symmetries of integrable systems of nonlinear water wave equations,such as dispersive water wave system, Broer-Kaup system and dispersiveless long wave system.Last section focuses on Benney system and its non-Noether symmetry, that appears to be local,gives rise to infinite sequence of conserved densities of Benney hierarchy and endows it withbi-Hamiltonian structure.

Regular Hamiltonian systems

The basic concept in geometric formulation of Hamiltonian dynamicsis notion of symplectic manifold. Such a manifold plays the role ofthe phase space of the dynamical system and therefore many propertiesof the dynamical system can be quite effectively investigated in the frameworkof symplectic geometry. Before we consider symmetries of the Hamiltonian dynamicalsystems, let us briefly recall some basic notions from symplectic geometry.
The symplectic manifold is a pair (M, ω)where M is smooth even dimensional manifold and ωis a closeddω = 0and nondegenerate 2-form on M. Being nondegenerate means thatcontraction of arbitrary non-zero vector field with ω does not vanishiXω = 0 ⇔ X = 0(here iX denotes contraction of the vector field Xwith differential form). Otherwise one can say that ωis nondegenerate if its n-th outer power does not vanish(ωn ≠ 0) anywhere on M.In Hamiltonian dynamics M is usually phase space of classical dynamical systemwith finite numbers of degrees of freedom and the symplectic form ωis basic object that defines Poisson bracket structure, algebra of Hamiltonian vector fieldsand the form of Hamilton's equations.
The symplectic form ω naturally defines isomorphism between vector fieldsand differential 1-forms on M (in other words tangent bundle TMof symplectic manifold can be quite naturally identified withcotangent bundle T*M).The isomorphic map Φω from TM intoT*M is obtained by taking contractionof the vector field with ωΦω: X → − iXω(minus sign is the matter of convention). This isomorphism gives rise to natural classificationof vector fields. Namely, vector field Xh is said to be Hamiltonianif its image is exact 1-form or in other words if it satisfies Hamilton's equationiXhω + dh = 0for some function h on M.Similarly, vector field X is called locally Hamiltonian if it's image is closed 1-formiXω + u = 0,        du = 0
One of the nice features of locally Hamiltonian vector fields, known as Liouville's theorem,is that these vector fields preserve symplectic form ω.In other words Lie derivative of the symplectic form ωalong arbitrary locally Hamiltonian vector field vanishesLXω = 0 ⇔ iXω + du = 0,       du = 0Indeed, using Cartan's formula that expresses Lie derivative in terms of contraction andexterior derivativeLX = iXd + diXone getsLXω = iXdω + diXω =diXω(since dω = 0) but according to the definition of locally Hamiltonianvector fielddiXω = − du = 0So locally Hamiltonian vector fields preserve ω and vise versa,if vector field preserves symplectic form ω then it is locally Hamiltonian.
Clearly, Hamiltonian vector fields constitute subset of locally Hamiltonian ones sinceevery exact 1-form is also closed. Moreover one can notice that Hamiltonian vector fields formideal in algebra of locally Hamiltonian vector fields. This fact can be observed as follows.First of all for arbitrary couple of locally Hamiltonian vector fields X, Ywe have LXω = LYω = 0 andLXLYω − LYLXω= L[X , Y]ω = 0so locally Hamiltonian vector fields form Lie algebra (corresponding Lie bracket is ordinarycommutator of vector fields). Further it is clear that for arbitrary Hamiltonian vector fieldXh and locally Hamiltonian one Z one hasLZω = 0andiXhω + dh = 0that impliesLZ(iXhω + dh) = L[Z , Xh]ω + iXhLZω +dLZh= L[Z , Xh]ω + dLZh = 0thus commutator [Z , Xh] is Hamiltonian vector fieldXLZh,or in other words Hamiltonian vector fields form ideal in algebra of locallyHamiltonian vector fields.
Isomorphism Φω can be extended tohigher order vector fields and differential forms by linearity and multiplicativity.Namely,Φω(X ∧ Y) =Φω(X) ∧ Φω(Y)Since Φω is isomorphism, the symplectic form ωhas unique counter image W known as Poisson bivector field.Property dω = 0 together with non degeneracy implies that bivectorfield W is also nondegenerate (Wn ≠ 0) and satisfiescondition[W , W] = 0where bracket [ , ] known as Schouten bracket or supercommutator, is actuallygraded extension of ordinary commutator of vector fields to the case of multivector fields,and can be defined by linearity and derivation property[C1 ∧ C2 ∧ ... ∧ Cn ,S1 ∧ S2 ∧ ... ∧ Sn] = (− 1)p + q[Cp , Sq] ∧C1 ∧ C2 ∧ ... ∧ Ĉp ∧ ... ∧ Cn ∧ S1 ∧ S2 ∧ ... ∧ Ŝq ∧ ...∧ Snwhere over hat denotes omission of corresponding vector field.In terms of the bivector field W Liouville's theorem mentioned above can berewritten as follows[W(u) , W] = 0 ⇔ du = 0for each 1-form u. It follows from graded Jacoby identity satisfied by Schoutenbracket and property [W , W] = 0 satisfied by Poisson bivector field.
Being counter image of symplectic form, W gives rise to mapΦW, transforming differential 1-forms into vector fields,which is inverted to the map Φω and is defined byΦW: u → W(u);       ΦWΦω = idFurther we will often use these maps.
In Hamiltonian dynamical systems Poisson bivector field is geometric object thatunderlies definition of Poisson bracket — kind of Lie bracket on algebra ofsmooth real functions on phase space. In terms of bivector field WPoisson bracket is defined by{f , g} = W(df ∧ dg)The condition [W , W] = 0 satisfied by bivector field ensures thatfor every triple (f, g, h) of smoothfunctions on the phase space the Jacobi identity{f{g , h}} + {h{f , g}} + {g{h , f}} = 0.is satisfied.Interesting property of the Poisson bracket is that map from algebra of real smooth functionson phase space into algebra of Hamiltonian vector fields, defined by Poisson bivector fieldf → Xf = W(df)appears to be homomorphism of Lie algebras. In other words commutator of two vector fieldsassociated with two arbitrary functions reproduces vector field associated with Poissonbracket of these functions[Xf , Xg] = X{f , g}This property is consequence of the Liouville theorem and definition of Poisson bracket.Further we also need another useful property of Hamiltonian vector fields and Poisson bracket{f , g} = W(df ∧ dg) = ω(Xf ∧ Xg) =LXfg = − LXggit also follows from Liouville theoremand definition of Hamiltonian vector fields and Poisson brackets.
To define dynamics on M one has to specify time evolution of observables(smooth functions on M). In Hamiltonian dynamical systems time evolutionis governed by Hamilton's equationddtf = {h , f}where h is some fixed smooth function on the phase space called Hamiltonian.In local coordinate frame zb bivector field Whas the formW = Wbc ∂∂zb ∧ ∂∂zcand the Hamilton's equation rewritten in terms of local coordinates takes the formżb = Wbc∂h∂zb

Non-Noether symmetries

Now let us focus on symmetries of Hamilton's equation (24).Generally speaking, symmetries play very important role in Hamiltonian dynamicsdue to different reasons. They not only give rise to conservation laws butalso often provide very effective solutions to problems that otherwise would be difficultto solve. Here we consider special class of symmetries of Hamilton's equationcalled non-Noether symmetries. Such a symmetries appear to be closely related tomany geometric concepts used in Hamiltonian dynamics including bi-Hamiltonian structures,Frölicher-Nijenhuis operators, Lax pairs and bicomplexes.
Before we proceedlet us recall that each vector field E on the phase space generatesthe one-parameter continuous group of transformationsgz = ezLE (here L denotes Lie derivative)that acts on the observables as followsgz(f) = ezLE(f) =f + zLEf + ½(zLE)2f + ⋯Such a group of transformation is called symmetry of Hamilton's equation (24)if it commutes with time evolution operatorddt gz(f)= gz(ddtf)in terms of the vector fields this condition means that the generatorE of the group gz commutes with the vector fieldW(h) = {h , }, i. e.[E , W(h)] = 0. However we would like to consider more generalcase where E is time dependent vector field on phase space. In this case(30) should be replaced with∂∂tE = [E , W(h)].
Further one should distinguish between groups of symmetry transformations generated by Hamiltonian,locally Hamiltonian and non-Hamiltonian vector fields. First kind of symmetriesare known as Noether symmetries and are widely used in Hamiltonian dynamics due to theirtight connection with conservation laws. Second group of symmetries is rarely used. While third group of symmetries that further will be referredas non-Noether symmetries seems to play important role in integrability issues due totheir remarkable relationship with bi-Hamiltonian structures andFrölicher-Nijenhuis operators. Thus if in addition to (30) thevector field E does not preserve Poisson bivector field [E , W] ≠ 0 thengz is called non-Noether symmetry.
Now let us focus on non-Noether symmetries. We would like to show that the presence ofsuch a symmetry essentially enriches the geometry of the phase spaceand under the certain conditions can ensure integrability of the dynamical system.Before we proceed let us recall that the non-Noether symmetry leads to a number ofintegrals of motion. More precisely therelationship between non-Noether symmetries and the conservation laws is described bythe following theorem. This theorem was proposed by Lutzky in [51].Here it is reformulated in terms of Poisson bivector field.
theorem. Let (M , h) be regular Hamiltonian system on the 2n-dimensionalPoisson manifold M. Then, if the vector field E generatesnon-Noether symmetry, the functionsY(k) = Ŵk ∧ Wn − kWn           k = 1,2, ... nwhere Ŵ = [E , W], are integrals of motion.
proof. By the definitionŴk ∧ Wn − k = Y(k)Wn.(definition is correct since the space of 2n degree multivector fields on 2ndegree manifold is one dimensional).Let us take time derivative of this expression along the vector field W(h),ddtŴk ∧ Wn − k =(ddtY(k))Wn+ Y(k)[W(h) , Wn]ork(ddtŴ) ∧ Ŵk − 1 ∧ Wn − k+ (n − k)[W(h) , W] ∧ Ŵk ∧ Wn − k − 1 = (ddtY(k))Wn+ nY(k)[W(h) , W] ∧ Wn − 1but according to the Liouville theorem the Hamiltonian vector field preserves W i. e.ddtW = [W(h) , W] = 0hence, by taking into account thatddtE= ∂∂tE + [W(h) , E] = 0 we getddtŴ =ddt[E , W] = [ddtE , W] + [E[W(h) , W]] = 0.and as a result (35) yieldsddtY(k)Wn = 0but since the dynamical system is regular (Wn ≠ 0)we obtain that the functions Y(k) are integrals of motion.
remark. Instead of conserved quantitiesY(1) ... Y(n), thesolutions c1 ... cn of the secular equation(Ŵ − cW)n = 0can be associated with the generator of symmetry.By expanding expression (40) it is easy to verify that the conservation lawsY(k) can be expressed in terms of the integrals of motionc1 ... cn in the following wayY(k) = (n − k)! k!n! ∑ms > mt cm1cm2 ⋯ cmkNote also that conservation laws Y(k) can be also defined by means ofsymplectic form ω using the following formulaY(k) = (LEω)k ∧ ωn − kωn       k = 1,2, ... nConservation laws c1 ... cn can be also derived fromthe secular equation(LEω − cω)n = 0However all these expressions fail in case of infinite dimensional Hamiltonian systemswhere the volume formΩ = ωndoes not exist since n = ∞. But fortunately in these case one can define conservation laws usingalternative formulaC(k) = iWk(LEω)kas far as it involves only finite degree differential forms(LEω)k and well defined multivector fieldsWk.Note that in finite dimensional case the sequence of conservation laws C(k)is related to families of conservation laws Y(k) and ck in thefollowing wayC(k) =(n − k)! k!n! ∑ms > mt cm1cm2 ⋯ cmk= n!(n − k)! k! Y(k)Note also that by taking Lie derivative of known conservation along the generator ofsymmetry E one can construct new conservation lawsddtY = LXhY = 0 ⇒ ddtLEY = LXhLEY =LELXhY = 0since [E , Xh] = 0.
remark. Besides continuous non-Noether symmetries generated by non-Hamiltonianvector fields one may encounter discrete non-Noether symmetries — noncannonicaltransformations that doesn't necessarily form group but commute with evolution operatorddt g(f) = g(ddtf)Such a symmetries give rise to the same conservation lawsY(k) = g(W)k ∧ Wn − kWn       k = 1,2, ... n
example. Let M be R4 with coordinatesz1, z2, z3, z4 and Poisson bivector fieldW =∂∂z1 ∧ ∂∂z3 +∂∂z2 ∧ ∂∂z4and let's take the following Hamiltonianh =12z12 +12z22 + ez3 − z4This is so called two particle non periodic Toda model.One can check that the vector field defined asE = 4∑s = 1 Es∂∂zswith componentsE1 =12z12 − ez3 − z4 −t2(z1 + z2)ez3 − z4E2 =12z22 + 2ez3 − z4 +t2(z1 + z2)ez3 − z4E3 =2z1 +12z2 +t2(z12 + ez3 − z4)E4 = z2 − 12z1 +t2(z22 + ez3 − z4)satisfies (31) condition and as a result generates symmetry of the dynamical system.The symmetry appears to be non-Noether with Schouten bracket [E , W] equal toŴ = [E , W] = z1∂∂z1 ∧ ∂∂z3+ z2∂∂z2 ∧ ∂∂z4+ ez3 − z4 ∂∂z1 ∧ ∂∂z2 +∂∂z3 ∧ ∂∂z4Calculation of volume vector fieldsŴk ∧ Wn − k gives rise toW ∧ W = − 2 ∂∂z1 ∧ ∂∂z2 ∧ ∂∂z3 ∧ ∂∂z4Ŵ ∧ W = − (z1 + z2)∂∂z1 ∧ ∂∂z2 ∧ ∂∂z3 ∧ ∂∂z4Ŵ ∧ Ŵ =− 2(z1z2 − ez3 − z4)∂∂z1 ∧ ∂∂z2 ∧ ∂∂z3 ∧ ∂∂z4and the conservation laws associated with this symmetry are justY(1) = Ŵ ∧ WW ∧ W =12(z1 + z2)Y(2) = Ŵ ∧ ŴW ∧ W =z1z2 − ez3 − z4It is remarkable that the same symmetry is also present in higher dimensions.For example in case where M is R6 with coordinatesz1, z2, z3, z4, z5, z6Poisson bivector equal toW = ∂∂z1 ∧ ∂∂z4 + ∂∂z2 ∧ ∂∂z5 +∂∂z3 ∧ ∂∂z6and the following Hamiltonianh =12z12 +12z22 +12z32 +ez4 − z5 +ez5 − z6we still can construct symmetry similar to (53).More precisely the vector field defined for arbitrary function F as E = 6∑s = 1 Es∂∂zswith components specified as followsE1 =12z12 − 2ez4 − z5 −t2(z1 + z2)ez4 − z5E2 =12z22 + 3ez4 − z5 −ez5 − z6 +t2(z1 + z2)ez4 − z5E3 =12z32 + 2ez5 − z6 +t2(z2 + z3)ez5 − z6E4 =3z1 + 12z2 + 12z3 +t2(z12 + ez4 − z5)E5 =2z2 − 12z1 + 12z3 +t2(z22 + ez4 − z5 +ez5 − z6)E6 = z3 − 12z1 − 12z2 +t2(z32 + ez5 − z6)satisfies (31) condition and generates non-Noether symmetry of the dynamical system(three particle non periodic Toda chain).Calculation of Schouten bracket [E , W] gives rise to expressionŴ = [E , W] = z1∂∂z1 ∧ ∂∂z4 +z2∂∂z2 ∧ ∂∂z5 +z3∂∂z3 ∧ ∂∂z6+ ez4 − z5 ∂∂z1 ∧ ∂∂z2 +ez5 − z6 ∂∂z2 ∧ ∂∂z3 +∂∂z4 ∧ ∂∂z5 + ∂∂z5 ∧ ∂∂z6Volume multivector fieldsŴk ∧ Wn − k can be calculated in the mannersimilar to R4 case and give rise to the well known conservation laws ofthree particle Toda chain.Y(1) = 16(z1 + z2 + z3) =Ŵ ∧ W ∧ WW ∧ W ∧ WY(2) = 13(z1z2 + z1z3 + z2z3− ez4 − z5 − ez5 − z6)= Ŵ ∧ Ŵ ∧ WW ∧ W ∧ WY(3) = z1z2z3 −z3ez4 − z5 −z1ez5 − z6 = Ŵ ∧ Ŵ ∧ ŴW ∧ W ∧ W

Non-Liouville symmetries

Besides Hamiltonian dynamical systems that admit invariant symplectic formω, there are dynamical systems that either are not Hamiltonian oradmit Hamiltonian realization but explicit form of symplectic structure ωis unknown or too complex. However usually such a dynamical systems possess invariant volume formΩ which like symplectic form can be effectively used in construction ofconservation laws. Note that volume form for given manifold is arbitrary differential formof maximal degree (equal to the dimension of manifold).In case of regular Hamiltonian systems, n-th outer power of the symplectic form ωnaturally gives rise to the invariant volume form known as Liouville formΩ = ωnand sometimes it is easier to work with Ω rather then with symplectic form itself.In generic Liouville dynamical system time evolution is governed by equations of motionddtf = X(f)where X is some smooth vector field that preserves Liouville volume formΩddtΩ = LXΩ = 0Symmetry of equations of motion still can be defined by conditionddt gz(f)= gz(ddtf)that in terms of vector fields implies that generator of symmetry E shouldcommute with time evolution operator X[E , X] = 0Throughout this chapter symmetry will be called non-Liouville if it is not conformal symmetryof Ω, or in other words ifLEΩ ≠ cΩfor any constant c.Such a symmetries may be considered as analog of non-Noether symmetriesdefined in Hamiltonian systems and similarly to the Hamiltonian case one can tryto construct conservation laws by means of generator of symmetry Eand invariant differential form Ω. Namely we have the followingtheorem, which is reformulation of Hojman's theorem in terms of Liouville volume form.
theorem. Let (M, X, Ω) be Liouville dynamical system on the smoothmanifold M. Then, if the vector field E generatesnon-Liouville symmetry, the functionJ = LEΩΩis conservation law.
proof. By the definitionLEΩ = JΩ.and J is not just constant(again definition is correct since the space of volume forms is one dimensional).By taking Lie derivative of this expression along the vector field X thatdefines time evolution we getLXLEΩ = L[X , E]Ω + LELXΩ = LX(JΩ) = (LXJ)Ω + JLXΩbut since Liouville volume form is invariant LXΩ = 0 andvector field E is generator of symmetry satisfying [E , X] = 0commutation relation we obtain(LXJ)Ω = 0orddtJ = LXJ = 0
remark. In fact theorem is valid for larger class of symmetries. Namely one can considersymmetries with time dependent generators. Note however that in this case condition[E , X] = 0 should be replaced by∂∂tE = [E , X]Note also that by calculating Lie derivative of conservation law J alonggenerator of the symmetry E one can recover additional conservation lawsJ(m) = (LE)mΩ
example. Let us consider symmetry of three particle non periodic Toda chain. This dynamical systemwith equations of motion ż4 = z1ż5 = z2ż6 = z3ż1 = − ez4 − z5ż2 = ez4 − z5 − ez5 − z6ż3 = ez5 − z6possesses invariant volume formΩ = dz1 ∧ dz2 ∧ dz3 ∧dz4 ∧ dz5 ∧ dz6The symmetry (61) is clearly non-Liouville one as far asLEΩ = (z1 + z2 + z3) Ωand main conservation law associated with this symmetry via Theorem 2 is total momentumJ = LEΩΩ = z1 + z2 + z3Other conservation laws can be recovered by taking Lie derivative of Jalong generator of symmetry E, in particularJ(1) = LEJ =12z12 +12z22 +12z32 +ez4 − z5 +ez5 − z6J(2) = LEJ(1) =12 (z13 + z23 + z33) + 32 (z1 + z2)ez4 − z5 +32 (z2 + z3)ez5 − z6

Lax Pairs

Presence of the non-Noether symmetry not only leads to a sequence of conservation laws, but alsoendows the phase space with a number of interesting geometric structures and it appears that such asymmetry is related to many important concepts used in theory of dynamical systems.One of the such concepts is Lax pair that plays quite important role in constructionof completely integrable models.Let us recall that Lax pair of Hamiltonian system on Poisson manifold M isa pair (L , P) of smooth functions on M with values in someLie algebra g such that the time evolution of L is given byadjoint actionddtL = [L , P] = − adPLwhere [ , ] is a Lie bracket on g. It is well known that each Laxpair leads to a number of conservation laws. When g is some matrix Lie algebrathe conservation laws are just traces of powers of LI(k) =12Tr(Lk)since trace is invariant under coadjoint actionddtI(k) = 12 ddt Tr(Lk) =12 Tr(ddtLk) = k2 Tr(Lk − 1ddtL) = k2 Tr(Lk − 1[L , P]) = 12 Tr([Lk, P]) = 0It is remarkable that each generator of the non-Noethersymmetry canonically leads to the Lax pair of a certain type.Such a Lax pairs have definite geometric origin, their Lax matrices are formedby coefficients of invariant tangent valued 1-form on the phase space.In the local coordinates zs, where the bivector fieldW, symplectic form ω and the generatorof the symmetry E have the following formW = ∑rs Wrs∂∂zr ∧ ∂∂zr      ω = ∑rs ωrsdzr ∧ dzs      E = ∑s Es∂∂zscorresponding Lax pair can be calculated explicitly.Namely we have the following theorem (see also [55]-[56]):
theorem. Let (M , h) be regular Hamiltonian system on the 2n-dimensionalPoisson manifold M.Then, if the vector field E on M generates the non-Noether symmetry,the following 2n×2n matrix valued functions on MLab = ∑dc ωadEc∂Wdb∂zc − Wbc∂Ed∂zc+ Wdc∂Eb∂zcPab = ∑c∂Wbc∂za·∂h∂zc + Wbc∂2h∂za∂zcform the Lax pair (84) of the dynamical system (M , h).
proof. Let us consider the following operator on a space of 1-formsŔE(u) = Φω([E , ΦW(u)]) − LEu(here ΦW and Φωare maps induced by Poisson bivector field and symplectic form).It is remarkable that ŔE appears to be invariant linear operator.First of all let us show that ŔE is really linear,or in other words, that for arbitrary 1-forms u and vand function f operator ŔE has the following propertiesŔE(u + v) = ŔE(u) + ŔE(v)andŔE(fu) = fŔE(u)First property is obvious consequence of linearity of Schouten bracket, Lie derivative andmaps ΦW, Φω.Second property can be checked directlyŔE(fu) = Φω([E , ΦW(fu)]) − LE(fu) = Φω([E , fΦW(u)]) − (LEf)u − fLEu = Φω((LEf)ΦW(u)) + Φω(f[E , ΦW(u)]) − (LEf)u − fLEu = LEfΦωΦW(u) + fΦω([E , ΦW(u)]) − (LEf)u − fLEu = f(Φω([E , ΦW(u)]) − LEu) = fŔE(u)as far as ΦωΦW(u) = u.Now let us check that ŔE is invariant operatorddtŔE = LXhŔE =LXh(ΦωLEΦW − LE)= ΦωL[Xh , E]ΦW− L[Xh, E] = 0because, being Hamiltonian vector field, Xh commutes with mapsΦW, Φω(this is consequence of Liouville theorem) and commutes with Eas far as E generates the symmetry [Xh, E] = 0.In the terms of the local coordinates ŔE has the following formŔE =∑abLab dza ⊗ ∂∂zband the invariance conditionddtŔE = LW(h)ŔE = 0yieldsddtŔE = ddt∑abLab dza ⊗ ∂∂zb= ∑ab (ddtLab) dza ⊗ ∂∂zb +∑ab Lab (LW(h)dza) ⊗ ∂∂zb+ ∑ab Lab dza ⊗ (LW(h)∂∂zb) =∑ab (ddtLab) dza ⊗ ∂∂zb+ ∑abcdLab∂Wad∂zc·∂h∂zd dzc ⊗ ∂∂zb+ ∑abcdLabWad∂2h∂zc∂zd dzc ⊗ ∂∂zb+ ∑abcdLab∂Wcd∂zb·∂h∂zddza ⊗ ∂∂zc + ∑abcd LabWcd ∂2h∂zb∂zd dza ⊗ ∂∂zc= ∑abddtLab + ∑c(PacLcb − LacPcb)dza ⊗ ∂∂zb = 0or in matrix notationsddtL = [L , P].So, we have proved that the non-Noether symmetry canonically yields a Lax pairon the algebra of linear operators on cotangent bundle over the phase space.
remark. The conservation laws (85)associated with the Lax pair (88) can be expressed in terms of theintegrals of motion ci in quite simple way:I(k) = 12 Tr(Lk) = ∑s cskThis correspondence follows from the equation (40)and the definition of the operator ŔE (89).One can also write down recursion relation that determines conservation lawsI(k) in terms of conservation laws C(k)I(m) + (− 1)mmC(m) +m − 1∑k = 1(− 1)k I(m − k)C(k) = 0
example. Let us calculate Lax matrix of two particle Toda chainassociated with non-Noether symmetry (53).Using (88) it is easy to check that Lax matrix has eight nonzero elementsL =z100− ez3 − z40z2ez3 − z4001z10− 100z2while matrix P involved in Lax pairddtL = [L , P]has the following formP =00100001− ez3 − z4ez3 − z400ez3 − z4− ez3 − z400The conservation laws associated with this Lax pairare total momentum and energy of two particle Toda chainI(1) = 12 Tr(L) = z1 + z2I(2) = 12 Tr(L2) = z12 + z22 + 2ez3 − z4Similarly one can construct Lax matrix of three particle Toda chain, it has 16 nonzero elementsL =z1000− ez4 − z500z20ez4 − z50− ez5 − z600z30ez5 − z600− 1− 1z10010− 10z2011000z3with non-zero elements matrix P listed belowP =000100000010000001− ez4 − z5ez4 − z50000ez4 − z5− ez4 − z5 − ez5 − z6ez5 − z60000ez5 − z6− ez5 − z6000Corresponding conservation laws reproduce total momentum, energy and secondHamiltonian involved in bi-Hamiltonian realization of Toda chainI(1) = 12 Tr(L) = z1 + z2I(2) = 12 Tr(L2) = z12 + z22 + z32 +2ez4 − z5 + 2ez5 − z6I(3) = 12 Tr(L3) =z13 + z23 + z33 +3(z1 + z2)ez4 − z5 +3(z2 + z3)ez5 − z6

Involutivity of conservation laws

Now let us focus on the integrability issues. We know thatn integrals of motion are associated with each generator of non-Noethersymmetry, in the same time we know that, according to the Liouville-Arnold theorem,regular Hamiltonian system (M, h) on 2n dimensional symplectic manifoldM is completely integrable (can be solved completely) if it admitsn functionally independent integrals of motion in involution.One can understand functional independence of set of conservation lawsc1, c2 ... cn aslinear independence of either differentials of conservation lawsdc1, dc2 ... dcn orcorresponding Hamiltonian vector fieldsXc1, Xc2 ... Xcn.Strictly speaking we can say that conservation laws c1, c2 ... cnare functionally independent if Lesbegue measure of the set of points of phase space Mwhere differentials dc1, dc2 ... dcn become linearly dependentis zero. Involutivity of conservation laws means that all possible Poisson brackets ofthese conservation laws vanish pair wise{ci , cj} = 0        i, j = 1... nIn terms of the vector fields, existence of involutive family of nfunctionally independent conservation lawsc1, c2 ... cnimplies that corresponding Hamiltonian vector fieldsXc1, Xc2 ... Xcnspan Lagrangian subspace (isotropic subspace of dimension n)of tangent space (at each point of M).Indeed, due to property (23){ci , cj} = ω(Xci , Xcj) = 0thus space spanned by Xc1, Xc2 ... Xcnis isotropic. Dimension of this space is n so it is Lagrangian. Note also that distributionXc1, Xc2 ... Xcnis integrable since due to (22)[Xci , Xcj] = X{ci , cj} = 0and according to Frobenius theorem there exists submanifold of M such thatdistribution Xc1, Xc2 ... Xcn spans tangentspace of this submanifold. Thus for phase space geometry existence of complete involutive setof integrals of motion implies existence of invariant Lagrangian submanifold.
Now let us look at conservation laws Y(1), Y(2) ... Y(n)associated with generator of non-Noether symmetry. Generally speaking these conservation laws might appear to be neither functionally independent nor involutive.However it is reasonable to ask the question – what condition should be satisfiedby the generator of the non-Noether symmetry to ensure the involutivity({Y(k) , Y(m)} = 0) of conserved quantities?In Lax theory situation is very similar — each Lax matrix leads to the set ofconservation laws but in general this set is not involutive, however in Lax theorythere is certain condition known as Classical Yang-Baxter Equation (CYBE)that being satisfied by Lax matrix ensures that conservation laws are in involution.Since involutivity of the conservation laws is closely related to the integrability,it is essential to have some analog of CYBE for the generatorof non-Noether symmetry. To address this issue we would like to propose the following theorem.
theorem. If the vector field E on 2n-dimensionalPoisson manifold M satisfies the condition[[E[E , W]]W] = 0and W bivector field has maximal rank (Wn ≠ 0)then the functions (32) are in involution{Y(k) , Y(m)} = 0
proof. First of all let us note thatthe identity (15) satisfied by the Poissonbivector field W is responsible for the Liouville theorem[W , W] = 0       ⇔       LW(f)W = [W(f) , W] = 0that follows from the graded Jacoby identity satisfied by Schouten bracket.By taking the Lie derivative of the expression (15)we obtain another useful identityLE[W , W] = [E[W , W]] = [[E , W] W] + [W[E , W]]= 2[Ŵ , W] = 0.This identity gives rise to the following relation[Ŵ , W] = 0      ⇔      [Ŵ(f) , W] = − [Ŵ , W(f)]and finally condition (110) ensures third identity[Ŵ , Ŵ] = 0yielding Liouville theorem for Ŵ[Ŵ , Ŵ] = 0      ⇔      [Ŵ(f) , Ŵ] = 0Indeed[Ŵ , Ŵ] = [[E , W]Ŵ] = [[Ŵ , E]W] = − [[E , Ŵ]W] = − [[E[E , W]]W] = 0Now let us consider two different solutions ci ≠ cjof the equation (40). By taking the Lie derivative of the equation(Ŵ − ciW)n = 0along the vector fields W(cj) andŴ(cj) and using Liouville theorem forW and Ŵ bivectors we obtain the following relations(Ŵ −ciW)n − 1(LW(cj)Ŵ− {cj , ci}W) =0,and(Ŵ −ciW)n − 1(ciLŴ(cj)W+ {cj , ci}∗W) = 0,where{ci , cj}∗ =Ŵ(dci ∧ dcj)is the Poisson bracket calculated by means of the bivector field Ŵ.Now multiplying (119) by ci subtracting (120) and usingidentity (114) gives rise to({ci , cj}∗ −ci{ci , cj})(Ŵ −ciW)n − 1W = 0Thus, either{ci , cj}∗ −ci{ci , cj} = 0or the volume field(Ŵ − ciW)n − 1Wvanishes. In the second case we can repeat(119)-(122) procedure forthe volume field(Ŵ − ciW)n − 1Wyielding after niterations Wn = 0 that according to ourassumption (that the dynamical system is regular) is not true.As a result we arrived at (123) and by the simpleinterchange of indices i ↔ j we get{ci , cj}∗ −cj{ci , cj} = 0Finally by comparing (123) and (124) we obtain thatthe functions ci are in involution with respect to the bothPoisson structures (since ci ≠ cj){ci , cj}∗ ={ci , cj} = 0and according to (41) the same is true for the integrals of motionY(k).
remark. Theorem 4 is useful in multidimensional dynamical systems where involutivity ofconservation laws can not be checked directly.

Bi-Hamiltonian systems

Further we will focus on non-Noether symmetries that satisfy condition (110). Besidesyielding involutive families of conservation laws, such a symmetries appear to be relatedto many known geometric structures such as bi-Hamiltonian systems [53]and Frölicher-Nijenhuis operators (torsionless tangent valued differential 1-forms).The relationship between non-Noether symmetries and bi-Hamiltonian structures wasalready implicitly outlined in the proof of Theorem 4. Now let us pay more attention tothis issue.
Originally bi-Hamiltonian structures were introduced by F. Magri in analisys ofintegrable infinite dimensional Hamiltonian systems such as Korteweg-de Vries (KdV) andmodified Korteweg-de Vries (mKdV) hierarchies, Nonlinear Schrödinger equationand Harry Dym equation. Since that time bi-Hamiltonian formalism is effectively usedin construction of involutive families of conservation laws in integrable models
Generic bi-Hamiltonian structure on 2n dimensional manifold consists outof two Poisson bivector fields W and Ŵ satisfying certaincompatibility condition [Ŵ , W] = 0. If, in addition, one of these bivectorfields is nondegenerate (Wn ≠ 0) then bi-Hamiltonian systemis called regular. Further we will discuss only regular bi-Hamiltonian systems.Note that each Poisson bivector field by definition satisfies condition (15). So we actuallyimpose four restrictions on bivector fields W and Ŵ[W , W] = [Ŵ , W] = [Ŵ , Ŵ] = 0andWn ≠ 0During the proof of Theorem 4 we already showed that bivector fieldsW and Ŵ = [E , W] satisfy conditions (126)(see (112)-(116)), thus we can formulate the following statement
theorem. Let (M , h) be regular Hamiltonian system on the 2n-dimensionalmanifold M endowed with regular Poisson bivector field W.Then, if the vector field E on M generates the non-Noether symmetry,and satisfies condition[[E[E , W]]W] = 0,the following bivector fields on MW, Ŵ = [E , W]form invariant bi-Hamiltonian system([W , W] = [Ŵ , W] = [Ŵ , Ŵ] = 0).
proof. See proof of Theorem 4.
remark. Bi-Hamiltonian systems obtained by taking Lie derivative of Poisson bivector along some vector field were studied in [70]
example. One can check that the non-Noether symmetry (53) satisfiescondition (110) while bivector fieldsW =∂∂z1 ∧ ∂∂z3 +∂∂z2 ∧ ∂∂z4andŴ = [E , W] = z1∂∂z1 ∧ ∂∂z3+ z2∂∂z2 ∧ ∂∂z4+ ez3 − z4 ∂∂z1 ∧ ∂∂z2 +∂∂z3 ∧ ∂∂z4form bi-Hamiltonian system [W , W] = [W , Ŵ] = [Ŵ , Ŵ] = 0.Similarly, one can recover bi-Hamiltonian system of three particle Toda chain associatedwith symmetry (61). It is formed by bivector fieldsW = ∂∂z1 ∧ ∂∂z4 + ∂∂z2 ∧ ∂∂z5 +∂∂z3 ∧ ∂∂z6andŴ = [E , W] = z1∂∂z1 ∧ ∂∂z4 +z2∂∂z2 ∧ ∂∂z5 +z3∂∂z3 ∧ ∂∂z6 + ez4 − z5 ∂∂z1 ∧ ∂∂z2 +ez5 − z6 ∂∂z2 ∧ ∂∂z3 +∂∂z4 ∧ ∂∂z5 + ∂∂z5 ∧ ∂∂z6
In terms of differential forms bi-Hamiltonian structure is formed by couple ofclosed differential 2-forms: symplectic form ω(such that dω = 0 and ωn ≠ 0)and ω∗ = LEω(clearly dω∗ = dLEω= LEdω = 0). It is important that by taking Lie derivative ofHamilton's equationiXhω + dh = 0along the generator E of symmetryLE(iXhω + dh) =i[E , Xh]ω + iXhLEω + LEdh =iXhω∗ + dLEh =0one obtains another Hamilton's equationiXhω∗ + dh∗ = 0where h∗ = LEh. This is actually second Hamiltonian realizationof equations of motion and thus under certain conditions existence of non-Noether symmetrygives rise to additional presymplectic structure ω∗and additional Hamiltonian realization of the dynamical system.In many integrable models admitting bi-Hamiltonian realization (including Toda chain,Korteweg-de Vries hierarchy, Nonlinear Schrödinger equation, Broer-Kaup system andBenney system) non-Noether symmetries that are responsible for existence of bi-Hamiltonian structureshas been found and motivated further investigation of relationship betweensymmetries and bi-Hamiltonian structures. Namely it seems to be interesting to knowwhether in general case existence of bi-Hamiltonian structure is related to non-Noether symmetry.Let us consider more general case and suppose that we have couple of differential 2-formsω and ω∗such thatdω = dω∗ = 0,       ωn ≠ 0iXhω + dh = 0andiXhω∗ + dh∗ = 0The question is whether there exists vector field E (generator of non-Noether symmetry)such that [E , Xh] = 0 andω∗ = LEω.
The answer depends on ω∗.Namely if ω∗ is exact form(there exists 1-form θ∗ such thatω∗ = dθ∗)then one can argue that such a vector field exists and thus anyexact bi-Hamiltonian structure is related to hidden non-Noethersymmetry. To outline proof of this statement let us introducevector field E∗ defined byiE∗ω = θ∗(such a vector field always exist because ωis nondegenerate 2-form).By constructionLE∗ ω = ω∗IndeedLE∗ω = diE∗ω +iE∗dω = dθ∗ = ω∗Andi[E∗, Xh]ω =LE∗(iXhω)− iXhLE∗ω =− d(E∗(h)− h∗) = − dh'In other words [Xh , E∗] is Hamiltonian vector field[Xh , E] = Xh'One can also construct locally Hamiltonian vector field Xg,that satisfies the same commutation relation. Namely let us definefunction (in general case this can be done only locally)g(z) = t∫0 h'dtwhere integration along solution of Hamilton's equation, with fixed origin and end point inz(t) = z, is assumed.And then it is easy to verify that locally Hamiltonian vector field associated with g(z),by construction, satisfies the same commutation relations asE∗ (namely [Xh , Xg] = Xh').Using E∗ and Xh'one can construct generator of non-Noether symmetry —non-Hamiltonian vector field E = E∗ − Xgcommuting with Xh and satisfyingLEω = LE∗ω− LXgω = LE∗ω = ω∗(thanks to Liouville's theorem LXgω = 0). So incase of regular Hamiltonian system every exact bi-Hamiltonian structure isnaturally associated with some (non-Noether) symmetry of space of solutions.In case where bi-Hamiltonian structure is not exact(ω∗ is closed but not exact) then due toω∗ = LEω =diEω + iEdω = diEωit is clear that such a bi-Hamiltonian system is not related to symmetry.However in all known cases bi-Hamiltonian structures seem to be exact.

Bidifferential calculi

Another important concept that is often used in theory of dynamical systems and maybe related to the non-Noether symmetry is the bidifferential calculus (bicomplex approach).Recently A. Dimakis and F. Müller-Hoissenapplied bidifferential calculi to the wide range of integrable modelsincluding KdV hierarchy, KP equation, self-dual Yang-Mills equation,Sine-Gordon equation, Toda models, non-linear Schrödingerand Liouville equations. It turns out that these models can be effectivelydescribed and analyzed using the bidifferential calculi[17], [24].Here we would like to show that each generator of non-Noether symmetrysatisfying condition [[E[E , W]]W] = 0 gives rise to certainbidifferential calculus.
Before we proceed let us specify what kind of bidifferential calculi we plan to consider.Under the bidifferential calculus we mean the graded algebra of differential formsover the phase spaceΩ =∞∪k = 0Ω(k)(Ω(k) denotes the space of k-degree differential forms)equipped with a couple of differential operatorsd, đ : Ω(k) → Ω(k + 1)satisfying conditionsd2 = đ2 = dđ + đd = 0 (see [24]). In other words we have two De Rhamcomplexes M, Ω, d and M, Ω, đon algebra of differential forms over the phase space. And these complexes satisfycertain compatibility condition — their differentials anticommute with each otherdđ + đd = 0.Now let us focus on non-Noether symmetries.It is interesting that if generator of the non-Noether symmetry satisfiesequation [[E[E , W]]W] = 0 then we are able to construct an invariantbidifferential calculus of a certain type.This construction is summarized in the following theorem:
theorem. Let (M , h) be regular Hamiltonian system on the Poisson manifold M.Then, if the vector field E on M generates the non-Noether symmetryand satisfies the equation[[E[E , W]]W] = 0,the differential operatorsdu =Φω([W , ΦW(u)])đu =Φω([[E , W]ΦW(u)])form invariant bidifferential calculus(d2 = đ2 = dđ + đd = 0)over the graded algebra of differential forms on M.
proof. First of all we have to show that d and đare really differential operators , i.e., they are linear maps fromΩ(k) intoΩ(k + 1), satisfy derivation property andare nilpotent (d2 = đ2 = 0).Linearity is obvious and follows from the linearity of the Schouten bracket [ , ]and ΦW, Φωmaps. Then, if u is a k-degree formΦW maps it on k-degree multivector field andthe Schouten brackets [W , ΦW(u)] and[[E , W]ΦW(u)] result thek + 1-degree multivector fields that are mapped on k + 1-degreedifferential forms by Φω.So, d and đare linear maps from Ω(k) intoΩ(k + 1).Derivation property follows from the same feature of the Schouten bracket[ , ] and linearity ofΦW andΦω maps.Now we have to prove the nilpotency of d and đ.Let us consider d2ud2u =Φω([W , ΦW(Φω([W , ΦW(u)]))])= Φω([W[W , ΦW(u)]]) = 0as a result of the property (112) and the Jacoby identity for [ , ] bracket.In the same mannerđ2u =Φω([[W , E][[W , E]ΦW(u)]]) = 0according to the property (116) of[W , E] = Ŵ and the Jacoby identity.Thus, we have proved that d and đ are differential operators(in fact d is ordinary exterior differential and the expression(151) is its well known representation in terms of Poisson bivector field).It remains to show that the compatibility condition dđ + đd = 0is fulfilled. Using definitions of d, đ and the Jacoby identity we get(dđ + đd)(u) =Φω([[[W , E]W]ΦW(u)]) = 0as far as (114) is satisfied.So, d and đ form the bidifferential calculus over the gradedalgebra of differential forms.It is also clear that the bidifferential calculus d, đis invariant, since both d and đ commute with time evolutionoperator W(h) = {h, }.
remark. Conservation laws that are associated with the bidifferential calculus(151) (152) and form Lenard scheme (see [24]): (k + 1)đI(k) = kdI(k + 1)coincide with the sequence of integrals of motion (98).Proof of this correspondence lays outside the scope of present manuscript,but can be done in the manner similar to [17].
example. The symmetry (53) endows R4 with bicomplex structured, đ where d is ordinary exterior derivative while đis defined byđz1 = z1dz1 − ez3 − z4dz4đz2 = z2dz2 + ez3 − z4dz3đz3 = z1dz3 + dz2đz4 = z2dz4 − dz1and is extended to whole De Rham complex by linearity, derivation property andcompatibility property dđ + đd = 0.By direct calculations one can verify that calculus constructed in this wayis consistent and satisfies đ2 = 0 property.To illustrate technique let us explicitly check that đ2z1 = 0.Indeedđ2z1 = đđz1 =đ(z1dz1 − ez3 − z4dz4) = đz1 ∧ dz1 + z1đdz1− ez3 − z4đz3 ∧ dz4+ ez3 − z4đz4 ∧ dz4− ez3 − z4đdz4 = đz1 ∧ dz1 − z1dđz1− ez3 − z4đz3 ∧ dz4+ ez3 − z4đz4 ∧ dz4+ ez3 − z4dđz4 = 0Because of propertiesđz1 ∧ dz1 =ez3 − z4dz1 ∧ dz4,− z1dđz1 =z1ez3 − z4dz3 ∧ dz4,− ez3 − z4đz3 ∧ dz4 = − z1ez3 − z4dz1 ∧ dz4− ez3 − z4dz2 ∧ dz4,ez3 − z4đz4 ∧ dz4 =ez3 − z4dz2 ∧ dz4andez3 − z4dđz4 =− ez3 − z4dz1 ∧ dz4Similarly one can show thatđ2z2 = đ2z3 = đ2z4 = 0and thus đ is nilpotent operator đ2 = 0.Note also that conservation lawsI(1) = z1 + z2I(2) = z12 + z22 + 2ez3 − z4form the simplest Lenard scheme2đI(1) = dI(2)Similarly one can construct bidifferential calculus associated with non-Noethersymmetry (61) of three particle Toda chain. In this case đcan be defined byđz1 = z1dz1 − ez4 − z5dz5đz2 = z2dz2 + ez4 − z5dz4− ez5 − z6dz6đz3 = z3dz3 + ez5 − z6dz5đz4 = z1dz4 − dz2 − dz3đz5 = z2dz5 + dz1 − dz3đz6 = z3dz6 + dz1 + dz2and as in case of two particle Toda itcan be extended to whole De Rham complex by linearity, derivation property andcompatibility property dđ + đd = 0.One can check that conservation laws of Toda chainI(1) = z1 + z2I(2) =z12 + z22 + z32 +2ez4 − z5 + 2ez5 − z6I(3) =z13 + z23 + z33 +3(z1 + z2)ez4 − z5 +3(z2 + z3)ez5 − z6form Lenard scheme2đI(1) = dI(2)3đI(2) = 2dI(3)

Frölicher-Nijenhuis geometry

Finally we would like to reveal some features of the operatorŔE(89) and to show how Frölicher-Nijenhuis geometry arises inHamiltonian system that possesses certain non-Noether symmetry.From the geometric properties of the tangent valued forms we knowthat the traces of powers of a linear operator Fon tangent bundle are in involution whenever its Frölicher-Nijenhuis torsionT(F) vanishes, i. e. whenever for arbitrary vector fields X,Y the conditionT(F)(X , Y) = [FX , FY] −F([FX , Y] + [X , FY] − F[X , Y]) = 0is satisfied.Torsionless forms are also called Frölicher-Nijenhuis operators and are widely used intheory of integrable models, where they play role of recursion operators and are usedin construction of involutive family of conservation laws.We would like to show that each generator of non-Noether symmetry satisfying equation[[E[E , W]]W] = 0canonically leads to invariant Frölicher-Nijenhuis operator on tangentbundle over the phase space. This operator can be expressed in terms of generator of symmetryand isomorphism defined by Poisson bivector field. Strictly speaking we have the following theorem.
theorem. Let (M , h) be regular Hamiltonian system on the Poisson manifold M.If the vector field E on M generates the non-Noether symmetryand satisfies the equation[[E[E , W]]W] = 0then the linear operator, defined forevery vector field X by equationRE(X) =ΦW(LEΦω(X))− [E , X]is invariant Frölicher-Nijenhuis operator on M.
proof. Invariance of RE follows from the invariance of theŔE defined by (89)(note that for arbitrary 1-form vector field u and vector field Xcontraction iXu has the propertyiREXu =iXŔEu,so RE is actually transposed toŔE).It remains to show that the condition (110) ensures vanishing of theFrölicher-Nijenhuis torsion T(RE) ofRE, i.e. for arbitrary vector fields X, Y we must getT(RE)(X , Y) = [RE(X) , RE(Y)] −RE([RE(X) , Y] + [X , RE(Y)] − RE([X , Y])) = 0First let us introduce the following auxiliary 2-formsω = Φω(W),       ω∗ = ŔEω       ω∗∗ = ŔEω∗Using the realization (151) of the differential dand the property (15) yieldsdω = Φω([W , W]) = 0Similarly, using the property (114) we obtaindω∗ =dΦω([E , W]) − dLEω =Φω([[E , W]W]) −LEdω = 0And finally, taking into account thatω∗ = 2Φω([E , W])and using the condition (110), we getdω∗∗ =2Φω([[E[E , W]]W])− 2dLEω∗ =− 2LEdω∗ = 0So the differential formsω, ω∗, ω∗∗are closeddω = dω∗ = dω∗∗ = 0Now let us consider the contraction of T(RE) and ω.iT(RE)(X , Y)ω =i[REX , REY]ω −i[REX , Y]ω∗ −i[X , REY]ω∗ +i[X , Y]ω∗∗=LREXiYω∗ −iREYLXω∗ −LREXiYω∗ +iYLREXω∗ −LXiREYω∗ +iREYLXω∗ +i[X , Y]ω∗∗= iYLXω∗∗ −LXiYω∗∗ +i[X , Y]ω∗∗ = 0where we used (175) (179),the property LXiYω =iYLXω + i[X , Y]ωof the Lie derivative and the relations of the following typeLREXω =diREXω + iREXdω= diXω∗= LXω∗ −iXdω∗ = LXω∗So we proved that for arbitrary vector fields X, Ythe contraction of T(RE)(X , Y) and ω vanishes.But since W bivector is non-degenerate(Wn ≠ 0), its counter imageω = Φω(W)is also non-degenerate and vanishing of the contraction (180)implies that the torsion T(RE) itself is zero.So we getT(RE)(X , Y) = [RE(X) , RE(Y)] −RE([RE(X) , Y] + [X , RE(Y)] − RE([X , Y])) = 0
example. The operator RE associated with non-Noethersymmetry (53) reproduces well known Frölicher-Nijenhuis operatorRE =z1dz1 ⊗ ∂∂z1 −dz1 ⊗ ∂∂z4 +z2dz2 ⊗ ∂∂z2 +dz2 ⊗ ∂∂z3 + z1dz3 ⊗ ∂∂z3 +ez3 − z4dz3 ⊗ ∂∂z2 +z2dz4 ⊗ ∂∂z4 −ez3 − z4dz4 ⊗ ∂∂z1(compare with [30]).The operator ŔEplays the role of recursion operator for conservation lawsI(1) = z1 + z2I(2) = z12 + z22 + 2ez3 − z4Indeed one can check that2ŔE(dI(1)) = dI(2)Similarly using non-Noether symmetry (61) one can construct recursion operator ofthree particle Toda chainRE = z1dz1 ⊗ ∂∂z1− ez4 − z5dz5 ⊗ ∂∂z1+ z2dz2 ⊗ ∂∂z2 +ez4 − z5dz4 ⊗ ∂∂z2− ez5 − z6dz6 ⊗ ∂∂z2+z3dz3 ⊗ ∂∂z3 +ez5 − z6dz5 ⊗ ∂∂z3+ z1dz4 ⊗ ∂∂z4− dz2 ⊗ ∂∂z4− dz3 ⊗ ∂∂z4+ z2dz5 ⊗ ∂∂z5 + dz1 ⊗ ∂∂z5− dz3 ⊗ ∂∂z5+ z3dz6 ⊗ ∂∂z6 +dz1 ⊗ ∂∂z6 + dz2 ⊗ ∂∂z6and as in case of two particle Toda chain, operator ŔEappears to be recursion operator for conservation lawsI(1) = z1 + z2I(2) =z12 + z22 + z32 +2ez4 − z5 + 2ez5 − z6I(3) =z13 + z23 + z33+ 3(z1 + z2)ez4 − z5 + 3(z2 + z3)ez5 − z6and fulfills the following recursion conditiondI(3) = 3ŔE(dI(2)) =6(ŔE)2(dI(1))

One-parameter families of conservation laws

One-parameter group of transformations gzdefined by (28) naturally acts on algebra of integrals of motion.Namely for each conservation lawddtJ = 0one can define one-parameter family of conserved quantities J(z)by applying group of transformations gz to JJ(z) = gz(J) = ezLEJ =J + zLEJ + ½(zLE)2J + ...Property (29) ensures that J(z) is conserved for arbitrary valuesof parameter zddtJ(z) =ddtgz(J) =gz(ddtJ) = 0and thus each conservation law gives rise to whole family of conservedquantities that form orbit of group of transformations ga.
Such an orbit J(z) is called involutive if conservation laws that formit are in involution{J(z1) , J(z2)} = 0(for arbitrary values of parameters z1, z2). On 2n dimensionalsymplectic manifold each involutive family that contains n functionally independentintegrals of motion naturally gives rise to integrable system (due to Liouville-Arnold theorem).So in order to identify those orbits that may be related to integrable models itis important to know how involutivity of family of conserved quantities J(z)is related to properties of initial conserved quantity J(0) = J and nature ofgenerator E of group gz = ezLE.In other words we would like to know what condition must be satisfied by generator ofsymmetry E and integral of motion J to ensure that{J(z1) , J(z2)} = 0. To address this issue and to describe class of vector fieldsthat possess nontrivial involutive orbits we would like to propose the followingtheorem
theorem. Let M be Poisson manifold endowed with 1-form ssuch that[W[W(s),W](s)] = c0[W(s)[W(s) ,W]]       (c0 ≠ − 1)Then each function J satisfying propertyW(LW(s)dJ) = c1[W(s),W](dJ)      (c1 ≠ 0)(c0,1 are some constants) gives rise to involutiveset of functionsJ(m) = (LW(s))mJ       {J(m), J(k)} = 0
proof. First let us inroduce linear operator R on bundle of multivector fields and define itfor arbitrary multivector field V by conditionR(V) = ½ ([W(s),V] − ΦW(LW(s)Φω(V)))Proof of linearity of this operator is identical to proof given for(89) so we will skip it. Further it is clear thatR(W) = [W(s),W]andR2(W) = R([W(s),W]) = ½([W(s)[W(s),W]] − ΦW((LW(s))2ω))= ½(1 + c0)[W(s)[W(s),W]]where we used propertyΦW((LW(s))2ω) =ΦW(LW(s)LW(s)ω) = ΦW(iW(s)dLW(s)ω) +ΦW(diW(s)LW(s)ω) = [W,ΦW(iW(s)LW(s)ω)] =[W[W(s),W](s)] = c0[W(s)[W(s),W]]At the same time by taking Lie derivative of(199) along the vector field W(s)one gets[W[W(s),W](s)] = (LW(s)R + R2)(W)comparing (200) and (202) yields(1 + c0)(LW(s)R + R2) = 2R2and thus(1 + c0)LW(s)R = (1 − c0)R2Further let us rewrite condition (196) as followsW(LW(s)dJ) = c1R(W)(dJ)due to linearity of operator R this condition can be extended toRm(W)(LW(s)dJ) = c1Rm + 1(W)(dJ)Now assuming that the following condition is trueW((LW(s))mdJ) = cmRm(W)(dJ)let us take its Lie derivative along vector field W(s).We getR(W)((LW(s))mdJ) + W((LW(s))m + 1dJ) = mcm1 − c01 + c0Rm + 1(W)(dJ) + cmRm + 1(W)(dJ)where we used properties (199) and (204).Note also that (207) together with linearity of operator Rimply thatRkW((LW(s))mdJ) = cmRk + m(W)(dJ)and thus (208) reduces toW((LW(s))m + 1dJ)= cm + 1Rm + 1(W)(dJ)where cm + 1 is defined by(1 + c0)cm + 1= mcn(1 − c0)So we proved that if assumtion (207) is valid for mthen it is also valid for m + 1, we also know that for m = 1 itmatches (205) and thus by induction we proved that condition(207) is valid for arbitrary m while cncan be determined bycm(1 + c0)m − 1 = c0(m − 1)!(1 − c0)m − 1Now using (207) and (209)it is easy to show that functions (LW(s))mJ are in involution.Indeed{(LW(s))mJ, (LW(s))kJ} =W(d(LW(s))mJ ∧ d(LW(s))kJ) = W((LW(s))mdJ ∧ (LW(s))kdJ) =cmckW(dJ ∧ dJ) = 0So we have proved that the functions (197) are in involution.
Further we will use this theorem to prove involutivity of family of conservation laws constructed using non-Noether symmetry of Toda chain.

Toda Model

To illustrate features of non-Noether symmetries we oftenrefer to two and three particle non-periodic Toda systems.However it turns out that non-Noether symmetries are present ingeneric n-particle non-periodic Toda chains as well, moreover they preservebasic features of symmetries (53), (61).In case of n-particle Toda model symmetry yields nfunctionally independent conservation laws in involution,gives rise to bi-Hamiltonian structure of Toda hierarchy,reproduces Lax pair of Toda system, endows phase space withFrölicher-Nijenhuis operator and leads to invariantbidifferential calculus on algebra of differential forms over phase spaceof Toda system.
First of all let us remind that Toda model is2n dimensional Hamiltonian system that describes the motionof n particles on the line governed by the exponential interaction.Equations of motion of the non periodic n-particle Toda model areddtqs = psddtps = ε(s − 1)eqs − 1 − qs −ε(n − s)eqs − qs + 1(ε(k) = − ε(− k) = 1 for any naturalk and ε(0) = 0) and can be rewritten in Hamiltonian form(24) with canonical Poisson bracket defined by Poisson bivectorW = n∑s = 1 ∂∂ps ∧ ∂∂qsand Hamiltonian equal toh = ½n∑s = 1ps2 +n − 1∑s = 1eqs − qs + 1Note that in two and three particle case we have used slightly different notationszs = pszn + s = qs        s = 1, 2, (3); n = 2(3)for local coordinates.The group of transformations gz generated by the vector fieldE will be symmetry of Toda chain if for eachps, qs satisfying Toda equations(214)gz(ps), gz(qs)also satisfy it.Substituting infinitesimal transformationsgz(ps) = ps + zE(ps) + O(z2)gz(ps) = qs + zE(qs) + O(z2)into (214) and grouping first order terms gives rise to theconditionsddtE(qs) = E(ps)ddtE(ps) = ε(s − 1)eqs − 1 − qs(E(qs − 1) − E(qs)) − ε(n − s)eqs − qs + 1(E(qs) − E(qs + 1))One can verify that the vector field defined byE(ps) = ½ps2 +ε(s − 1)(n − s + 2)eqs − 1 − qs −ε(n − s)(n − s) eqs − qs + 1+ t2(ε(s − 1)(ps − 1 + ps)eqs − 1 − qs −ε(n − s)(ps + ps + 1)eqs − qs + 1)E(qs) = (n − s + 1)ps −½s − 1∑k = 1 pk+ ½n∑k = s + 1 pk+ t2(ps2 +ε(s − 1)eqs − 1 − qs +ε(n − s)eqs − qs + 1)satisfies (31) and generates symmetry of Toda chain. It appears that this symmetry is non-Noether since it does notpreserve Poisson bracket structure [E , W] ≠ 0and additionally one can check that Yang-Baxter equation[[E[E , W]]W] = 0 is satisfied.This symmetry may play important role inanalysis of Toda model. First let us note that calculating LEWleads to the following Poisson bivector fieldŴ = [E , W] =n∑s = 1 ps∂∂ps ∧ ∂∂qs+ n − 1∑s = 1 eqs − qs + 1 ∂∂ps ∧ ∂∂qs + 1+ ∑r > s ∂∂qs ∧ ∂∂qrand together W and LEW give rise tobi-Hamiltonian structure of Toda model (compare with [30]).Thus bi-Hamiltonian realization of Toda chain can be considered as manifestationof hidden symmetry.In terms of bivector fields these bi-Hamiltonian system is formed byThe conservation laws (45) associated with the symmetry reproduce well knownset of conservation laws of Toda chain.I(1) = C(1) = n∑s = 1psI(2) = (C(1))2 − 2C(2) =n∑s = 1 ps2 + 2n − 1∑s = 1eqs − qs + 1I(3) = C(1))3 − 3C(1)C(2)+ 3C(3) = n∑s = 1 ps3 +3n − 1∑s = 1 (ps + ps + 1) eqs − qs + 1I(4) = C(1))4 − 4(C(1))2C(2) +2(C(2))2 + 4C(1)C(3) − 4C(4)= n∑s = 1 ps4 + 4n − 1∑s = 1(ps2 + 2psps + 1 + ps + 12)eqs − qs + 1+ 2n − 1∑s = 1 e2(qs − qs + 1) +4n − 2∑s = 1 eqs − qs + 2 I(m) = (− 1)m + 1mC(m) +m − 1∑k = 1(− 1)k + 1I(m − k)C(k)The condition [[E[E , W]]W] = 0 satisfied by generator of thesymmetry E ensures that the conservation laws are in involutioni. e. {C(k) , C(m)} = 0.Thus the conservation laws as well as the bi-Hamiltonian structureof the non periodic Toda chain appear to be associated with non-Noether symmetry.
Using formula (88) one can calculate Lax pairassociated with symmetry (220).Lax matrix calculated in this way has the following non-zero entries(note that in case of n = 2 and n = 3 this formula yields matrices(102)-(105))Lk, k = Ln + k, n + k = pkLn + k, k + 1 = − Ln + k + 1, k =ε(n − k)eqk − qk + 1Lk, n + m = ε(m − k)m, k = 1, 2, ... , nwhile non-zero entries of P matrix involved in Lax pair arePk, n + k = 1Pn + k, k = − ε(k − 1)eqk − 1 − qk− ε(n − k)eqk − qk + 1Pn + k, k + 1 = ε(n − k)eqk − qk + 1Pn + k, k − 1 = ε(k − 1)eqk − 1 − qkk = 1, 2, ... , nThis Lax pair constructed from generator of non-Noether symmetryexactly reproduces known Lax pair of Toda chain.
Like two and three particle Toda chain, n-particle Toda model also admitsinvariant bidifferential calculus on algebra of differential forms over the phase space.This bidifferential calculus can be constructed using non-Noether symmetry (see (152)),it consists out of two differential operators d, đwhere d is ordinary exterior derivative while đcan be defined byđqs = psdqs + ∑r > sdpr − ∑s > rdprđps = psdps − eqs − qs + 1dqs + 1+ eqs − 1 − qsdqsand is extended to whole De Rham complex by linearity, derivation property andcompatibility property dđ + đd = 0.By direct calculations one can verify that calculus constructed in this wayis consistent and satisfies đ2 = 0 property.One can also check that conservation laws (222) form Lenard scheme(k + 1)đI(k) = kdI(k + 1)
Further let us focus on Frölicher-Nijenhuis geometry. Using formula (173)one can construct invariant Frölicher-Nijenhuis operator, out of generator of non-Noethersymmetry of Toda chain. Operator constructed in this way has the formŔE = n∑s = 1ps(dps ⊗ ∂∂qs + dqs ⊗ ∂∂ps) − n − 1∑s = 1 eqs − qs + 1dqs + 1 ⊗ ∂∂ps+ n − 1∑s = 1 eqs − 1 − qsdqs ⊗ ∂∂ps− ∑s > r (dps ⊗ ∂∂qr − dpr ⊗ ∂∂qs)One can check that Frölicher-Nijenhuis torsion of this operator vanishes andit plays role of recursion operator for n-particle Toda chain in sense that conservation lawsI(k) satisfy recursion relation(k + 1)RE(dI(k)) = kdI(k + 1)Thus non-Noether symmetry of Toda chain not only leads ton functionally independent conservation laws in involution, but alsoessentially enriches phase space geometry by endowing it withinvariant Frölicher-Nijenhuis operator, bi-Hamiltonian system,bicomplex structure and Lax pair.
Finally, in order to outline possible applications of Theorem 8 let us studyaction of non-Noether symmetry (220) on conserved quantitiesof Toda chain. Vector field E defined by (220) generatesone-parameter group of transformations (28) that maps arbitraryconserved quantity J toJ(z) = J + zJ(1) + z22!J(2) +z33!J(3) + ⋯whereJ(m) = (LE)mJIn particular let us focus on family of conserved quantities obtained by action ofga = eaLE on total momenta of Toda chainJ = n∑s = 1 psBy direct calculations one can check that family J(z), that forms orbitof non-Noether symmetry generated by E, reproduces entire involutivefamily of integrals of motion (222). NamelyJ(1) = LEJ = ½ n∑s = 1 ps2 + n − 1∑s = 1eqs − qs + 1J(2) = LEJ(1) = (LE)2J =12 n∑s = 1ps3 +32n − 1∑s = 1 (ps + ps + 1)eqs − qs + 1J(3) = LEJ(2) = (LE)3J =¾ n∑s = 1ps4 +3n − 1∑s = 1(ps2 + 2psps + 1 +ps + 12)eqs − qs + 1+ 32 n − 1∑s = 1 e2(qs − qs + 1) +3n − 2∑s = 1 eqs − qs + 2J(m) = LEJ(m − 1) = (LE)mJ
Involutivity of this set of conservation laws can be verified using Theorem 8.In particular one can notice that differential 1-form s defined byE = W(s)(where E is generator of non-Noether symmetry (220))satisfies condition[W[W(s),W](s)] = 3[W(s)[W(s) ,W]]while conservation law J defined by (231)has propertyW(LW(s)dJ) = − [W(s),W](dJ)and thus according to Theorem 8 conservation laws (232)are in involution.

Korteweg-de Vries equation

Toda model provided good example of finite dimensional integrable Hamiltonian systemthat possesses non-Noether symmetry. However there are manyinfinite dimensional integrable Hamiltonian systems and in this case inorder to ensure integrability one should constructinfinite number of conservation laws. Fortunately in several integrable modelsthis task can be effectively simplified by identifying appropriate non-Noether symmetry.First let us consider well known infinite dimensional integrable Hamiltonian system –Korteweg-de Vries equation (KdV). The KdV equation has the following formut + uxxx + uux = 0(here u is smooth function of (t, x) ∈ R2).The generators of symmetries of KdV should satisfy conditionE(u)t + E(u)xxx +uxE(u) + uE(u)x = 0which is obtained by substituting infinitesimal transformationu → u + zE(u) + O(z2) into KdV equation and grouping first order terms.
Later we will focus on the symmetry generated by the following vector fieldE(u) = 2uxx + 23u2 + 16uxv +x2(uxxx + uux) − t4(6uxxxxx + 20uxuxx +10 uuxxx + 5u2ux)(here v is defined by vx = u).
If u is subjected to zero boundary conditions u(t, − ∞) = u(t, + ∞) = 0then KdV equation can be rewritten in Hamiltonian form ut = {h , u}with Poisson bivector field equal toW = + ∞∫− ∞dx δδu∧ {δδu}xand Hamiltonian defined byh = + ∞∫− ∞ (ux2 − u33) dxBy taking Lie derivative of thesymplectic form along the generator of the symmetry one getssecond Poisson bivector [E , W] = W = + ∞∫− ∞dx ({δδu}xx ∧ {δδu}x+ 23uδδu ∧ {δδu}x)involved in bi-Hamiltonian structure of KdV hierarchy andproposed by Magri [58].
Now let us show how non-Noether symmetry can be used to construct conservation lawsof KdV hierarchy. By integrating KdV it is easy to show thatJ(0) = + ∞∫− ∞u dxis conserved quantity. At the same time Lie derivative of any conservedquantity along generator of symmetry is conserved as well,while taking Lie derivative of J(0) along E gives rise toinfinite sequence of conservation laws J(m) = (LE)mJ(0)that reproduce well known conservation laws of KdV equationJ(0) = + ∞∫− ∞u dxJ(1) = LEJ(0) =¼+ ∞∫− ∞u2 dx J(2) = (LE)2J(0) =58+ ∞∫− ∞(u33 − ux2) dx J(3) = (LE)3J(0) = 3516+ ∞∫− ∞ (536u4 −53uux2 + uxx2) dxJ(m) = (LE)mJ(0)Thus the conservation laws and bi-Hamiltonian structures of KdV hierarchy are related to the non-Noether symmetry of KdV equation.

Nonlinear water wave equations

Among nonlinear partial differential equations that describe propagation of waves in shallow waterthere are many remarkable integrable systems. We have already discussed case of KdV equation, that possess non-Noether symmetries leading to the infinite sequence of conservation lawsand bi-Hamiltonian realization of these equations,now let us consider other important water wave systems.It is reasonable to start with dispersive water wave system [73],[74],since many other models can be obtained from it by reduction.Evolution of dispersive water wave system is governed bythe following set of equationsut = uxw + uwxvt = uux − vxx + 2vxw + 2vwxwt = wxx − 2vx + 2wwxEach symmetry of this system must satisfy linear equationE(u)t = (wE(u))x + (uE(w))xE(v)t = (uE(u))x − E(v)xx + 2(wE(v))x + 2(vE(w))xE(w)t = E(w)xx − 2E(v)x + 2(wE(w))xobtained by substituting infinitesimal transformationsu → u + zE(u) + O(z2)v → v + zE(v) + O(z2)w → w + zE(w) + O(z2)into equations of motion (245) and grouping first order(in a) terms. One of the solutions of this equation yieldsthe following symmetry of dispersive water wave systemE(u) = uw + x(uw)x + 2t(uw2 − 2uv + uwx)xE(v) = 32u2 + 4vw − 3vx + x(uux + 2(vw)x − vxx)+ 2t(u2w − uux − 3v2 + 3vw2 − 3vxw + vxx)xE(w) = w2 + 2wx − 4v + x(2wwx + wxx − 2vx)− 2t(u2 + 6vw − w3 − 3wwx − wxx)xand it is remarkable that this symmetry is local in sense that E(u) in pointx depends only on u and its derivatives evaluated in the same point,(this is not the case in KdV where symmetry is non localdue to presence of non local field v defined by vx = u).
Before we proceed let us note that dispersive water wave system is actually infinite dimensionalHamiltonian dynamical system. Assuming that u, v and w fieldsare subjected to zero boundary conditionsu(± ∞) = v(± ∞) = w(± ∞) = 0it is easy to verify that equations (245) can be represented in Hamiltonian formut = {h , u}vt = {h , v}wt = {h , w}with Hamiltonian equal toh = − ¼ + ∞∫− ∞ (u2w + 2vw2 − 2vxw − 2v2)dxand Poisson bracket defined by the following Poisson bivector fieldW = + ∞∫− ∞ {½ δδu ∧ {δδu}x +δδv ∧ {δδw}x} dxNow using our symmetry that appears to be non-Noether, one can calculate second Poissonbivector field involved in the bi-Hamiltonian realization of dispersive water wave systemŴ = [E , W] = − 2 + ∞∫− ∞{u δδv ∧ {δδu}x+ v δδv ∧ {δδv}x+ {δδv}x ∧ {δδw}x+ w δδv ∧ {δδw}x+ {δδw}x ∧ δδw} dxNote that Ŵ give rise to the second Hamiltonian realization ofthe modelut = {h∗ , u}∗vt = {h∗ , v}∗wt = {h∗ , w}∗whereh∗ = − ¼ + ∞∫− ∞ (u2 + 2vw)dxand { , }∗ is Poisson bracket defined bybivector field Ŵ.
Now let us pay attention to conservation laws. By integrating third equationof dispersive water wave system (245) it is easy to show thatJ(0) =+ ∞∫− ∞wdxis conservation law. Using non-Noether symmetryone can construct other conservation laws by taking Lie derivativeof J(0) along the generator of symmetry and in this wayentire infinite sequence of conservation laws of dispersive water wave systemcan be reproducedJ(0) = + ∞∫− ∞ wdxJ(1) = LEJ(0) = − 2 + ∞∫− ∞ vdx J(2) = LEJ(1) = (LE)2J(0) =− 2+ ∞∫− ∞ (u2 + 2vw)dxJ(3) = LEJ(2) = (LE)3J(0) =− 6+ ∞∫− ∞ (u2w + 2vw2 − 2vxw − 2v2)dxJ(4) = LEJ(3) = (LE)4J(0)= − 24 + ∞∫− ∞ (u2w2 + u2wx − 2u2v − 6v2w +2vw3 − 3vxw2 − 2vxwx)dxJ(n) = LEJ(n − 1) = (LE)nJ(0)Thus conservation laws and bi-Hamiltonian structure of dispersive waterwave system can be constructed by means of non-Noether symmetry.
Note that symmetry (248) can be used in many otherpartial differential equations that can be obtained by reduction from dispersivewater wave system. In particular one can use it in dispersiveless water wave system,Broer-Kaup system, dispersiveless long wave system, Burger's equation etc.In case of dispersiveless water waves systemut = uxw + uwxvt = uux + 2vxw + 2vwxwt = − 2vx + 2wwxsymmetry (248) is reduced toE(u) = uw + x(uw)x + 2t(uw2 − 2uv)xE(v) = 32u2 + 4vw + x(uux + 2(vw)x)+ 2t(u2w − 3v2 + 3vw2)xE(w) = w2 − 4v + x(2wwx − 2vx) − 2t(u2 + 6vw − w3)xand corresponding conservation laws (257) reduce toJ(0) = + ∞∫− ∞ wdxJ(1) = LEJ(0) =− 2 + ∞∫− ∞ vdxJ(2) = LEJ(1) = (LE)2J(0) =− 2 + ∞∫− ∞ (u2 + 2vw)dxJ(3) = LEJ(2) = (LE)3J(0) =− 6 + ∞∫− ∞ (u2w + 2vw2 − 2v2)dxJ(4) = LEJ(3) = (LE)4J(0) =− 24 + ∞∫− ∞ (u2w2 − 2u2v − 6v2w + 2vw3)dxJ(n) = LEJ(n − 1) = (LE)nJ(0)
Another important integrable model that can be obtained from dispersive water wave systemis Broer-Kaup system [73],[74]vt = ½ vxx + vxw + vwxwt = − ½ wxx + vx + wwxOne can check that symmetry (248) of dispersive water wave system,after reduction, reproduces non-Noether symmetry of Broer-Kaup modelE(v) = 4vw + 3vx + x(2(vw)x + vxx)+ t(3v2 + 3vw2 + 3vxw + vxx)xE(w) = w2 − 2wx + 4v + x(2wwx − wxx + 2vx)+ t(6vw + w3 − 3wwx + wxx)xand gives rise to the infinite sequence of conservation laws of Broer-Kaup hierarchyJ(0) = + ∞∫− ∞ wdxJ(1) = LEJ(0) = 2 + ∞∫− ∞ vdxJ(2) = LEJ(1) = (LE)2J(0) =4 + ∞∫− ∞ vwdxJ(3) = LEJ(2) = (LE)3J(0) =12 + ∞∫− ∞ (vw2 + vxw + v2)dxJ(4) = LEJ(3) = (LE)4J(0) =24 + ∞∫− ∞ (6v2w + 2vw3 + 3vxw2 − 2vxwx)dxJ(n) = LEJ(n − 1) = (LE)nJ(0)
And exactly like in the dispersive water wave system one can rewrite equations of motion(261) in Hamiltonian formvt = {h , v}wt = {h , w}where Hamiltonian ish = ½ + ∞∫− ∞ (vw2 + vxw + v2)dxwhile Poisson bracket is defined by the Poisson bivector fieldW = + ∞∫− ∞ {δδv ∧ {δδw}x} dxAnd again, using symmetry (262) one can recover second Poissonbivector field involved in the bi-Hamiltonian realization of Broer-Kaup systemby taking Lie derivative of (266)Ŵ = [E , W] = − 2 + ∞∫− ∞ {v δδv ∧ {δδv}x− {δδv}x ∧ {δδw}x + w δδv ∧ {δδw}x+ δδw ∧ {δδw}x} dxThis bivector field give rise to the second Hamiltonian realization ofthe Broer-Kaup systemvt = {h∗ , v}∗wt = {h∗ , w}∗withh∗ = −¼ + ∞∫− ∞vwdxSo the non-Noether symmetry of Broer-Kaup system yields infinite sequenceof conservation laws of Broer-Kaup hierarchy and endows it with bi-Hamiltonian structure.
By suppressing dispersive terms in Broer-Kaup system one reduces it to more simpleintegarble model — dispersiveless long wave system [73],[74]vt = vxw + vwxwt = vx + wwxin this case symmetry (248) reduces to more simple non-Noether symmetryE(v) = 4vw + 2x(vw)x + 3t(v2 + vw2)xE(w) = w2 + 4v + 2x(wwx + vx) + t(6vw + w3)xwhile the conservation laws of Broer-Kaup hierarchy reduce tosequence of conservation laws of dispersiveless long wave systemJ(0) = + ∞∫− ∞ wdxJ(1) = LEJ(0) = 2 + ∞∫− ∞ vdxJ(2) = LEJ(1) = (LE)2J(0) =4 + ∞∫− ∞ vwdxJ(3) = LEJ(2) = (LE)3J(0) =12 + ∞∫− ∞ (vw2 + v2)dxJ(4) = LEJ(3) = (LE)4J(0) =48 + ∞∫− ∞ (3v2w + vw3)dxJ(n) = LEJ(n − 1) = (LE)nJ(0)
At the same time bi-Hamitonian structure of Broer-Kaup hierarchy, after reductiongives rise to bi-Hamiltonian structure of dispersiveless long wave systemW = + ∞∫− ∞ {δδv ∧ {δδw}x} dxŴ = [E , W] = − 2 + ∞∫− ∞ {v δδv ∧ {δδv}x+ w δδv ∧ {δδw}x + δδw ∧ {δδw}x} dx
Among other reductions of dispersive water wave system one should probably mentionBurger's equation [73],[74]wt = wxx + wwxHowever Hamiltonian realization of this equation is unknown(for instance Poisson bivector field of dispersive water wave system(252) vanishes during reduction).

Benney system

Now let us consider another integrable system of nonlinear partialdifferential equations — Benney system [73],[74]. Time evolution of this dynamicalsystem is governed by equations of motion ut = vvx + 2(uw)xvt = 2ux + (vw)xwt = 2vx + 2wwxTo determine symmetries of the system one has to look for solutions oflinear equationE(u)t = (vE(v))x + 2(uE(w))x + 2(wE(u))xE(v)t = 2E(u)x + (vE(w))x + (wE(v))xE(w)t = 2E(v)x + 2(wE(w))xobtained by substituting infinitesimal transformationsu → u + zE(u) + O(z2)v → v + zE(v) + O(z2)w → w + zE(w) + O(z2)into equations (275) and grouping first order terms.In particular one can check that the vector field E defined byE(u) = 5uw + 2v2 + x(2(uw)x + vvx) + 2t(4uv + v2w + 3uw2)xE(v) = vw + 6u + x((vw)x + 2ux) + 2t(4uw + 3v2 + vw2)xE(w) = w2 + 4v + 2x(wwx + vx) + 2t(w3 + 4vw + 4u)xsatisfies equation (276) and therefore generates symmetry of Benney system.The fact that this symmetry is local simplifies further calculations.
At the same time, it is known fact, that under zero boundary conditionsu(± ∞) = v(± ∞) = w(± ∞) = 0Benney equations can be rewritten in Hamiltonian formut = {h , u}vt = {h , v}wt = {h , w}with Hamiltonianh = − ½ + ∞∫− ∞ (2uw2 + 4uv + v2w)dxand Poisson bracket defined by the following Poisson bivector fieldW = + ∞∫− ∞{½ δδv ∧ {δδv}x+ δδu ∧ {δδw}x} dxUsing symmetry (278) that in fact is non-Noether one, we can reproducesecond Poisson bivector field involved in the bi-Hamiltonian structure of Benney hierarchy(by taking Lie derivative of W along E)Ŵ = [E , W] = − 3 + ∞∫− ∞u δδu ∧ {δδu}x+ v δδu ∧ {δδv}x+ w δδu ∧ {δδw}x+ 2 δδv ∧ {δδw}x} dxPoisson bracket defined by bivector field Ŵ gives riseto the second Hamiltonian realization of Benney systemut = {h∗ , u}∗vt = {h∗ , v}∗wt = {h∗ , w}∗with new Hamiltonianh∗ = 16 + ∞∫− ∞ (v2 + 2uw)dxThus symmetry (278) is closely related tobi-Hamiltonian realization of Benney hierarchy.
The same symmetry yields infinite sequence of conservation laws of Benney system.Namely one can construct sequence of integrals of motion by applying non-Noethersymmetry (278) toJ(0) = + ∞∫− ∞wdx(the fact that J(0) is conserved can be verified by integratingthird equation of Benney system). The sequence looks likeJ(0) = + ∞∫− ∞ wdxJ(1) = LEJ(0) =2 + ∞∫− ∞ vdxJ(2) = LEJ(1) = (LE)2J(0) =8 + ∞∫− ∞ udxJ(3) = LEJ(2) = (LE)3J(0) =12 + ∞∫− ∞ (v2 + 2uw)dxJ(4) = LEJ(3) = (LE)4J(0) =48 + ∞∫− ∞ (2uw2 + 4uv + v2w)dxJ(5) = LEJ(4) = (LE)5J(0) =240 + ∞∫− ∞ (4u2 + 8uvw + 2uw3 + 2v3 + v2w2)dxJ(n) = LEJ(n − 1) = (LE)nJ(0)So conservation laws and bi-Hamiltonian structure of Benney hierarchyare closely related to its symmetry, that can play important role in analysis ofBenney system and other models that can be obtained from it by reduction.

Conclusions

The fact that many important integrable models, such as Korteweg-de Vries equation, Broer-Kaup system, Benney system and Toda chain,possess non-Noether symmetries that can be effectively usedin analysis of these models, inclines us to think that non-Noether symmetries can playessential role in theory of integrable systems and properties of this class of symmetriesshould be investigated further.The present review indicates that in many cases non-Noether symmetries lead to maximal involutivefamilies of functionally independent conserved quantities and in this way ensure integrabilityof dynamical system. To determine involutivity of conservation laws in cases when it can not be checkedby direct computations (for instance one can not check directly the involutivityin many generic n-dimensional models like Toda chainand infinite dimensional models like KdV hierarchy)we propose analog of Yang-Baxter equation, that being satisfied bygenerator of symmetry, ensures involutivity of familyof conserved quantities associated with this symmetry.
Another important feature of non-Noether symmetries is their relationship withseveral essential geometric concepts, emerging in theory of integrable systems, such asFrölicher-Nijenhuis operators, Lax pairs, bi-Hamiltonian structures andbicomplexes. On the one hand this relationship enlarges possible scope ofapplications of non-Noether symmetries in Hamiltonian dynamics and on the other hand itindicates that existence of invariant Frölicher-Nijenhuis operators,bi-Hamiltonian structures and bicomplexes in many cases can be considered as manifestationof hidden symmetries of dynamical system.
acknowledgements. Author is grateful to George Jorjadze, Zakaria Giunashvili andMichael Maziashvili for constructive discussions and help.This work was supported by INTAS (00-00561).

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