_{X}ω = 0 ⇔ X = 0

_{X}denotes contraction of the vector field $X$ with differential form). Otherwise one can say that $\omega $ is nondegenerate if its n-th outer power does not vanish ($\omega n\ne \; 0$) anywhere on $M$. In Hamiltonian dynamics $M$ is usually phase space of classical dynamical system with finite numbers of degrees of freedom and the symplectic form $\omega $ is basic object that defines Poisson bracket structure, algebra of Hamiltonian vector fields and the form of Hamilton's equations.

_{ω}from $TM$ into $T*M$ is obtained by taking contraction of the vector field with $\omega $

_{ω}: X → − i

_{X}ω

_{h}is said to be Hamiltonian if its image is exact 1-form or in other words if it satisfies Hamilton's equation

_{Xh}ω + dh = 0

_{X}ω + u = 0, du = 0

_{X}ω = 0 ⇔ i

_{X}ω + du = 0, du = 0

_{X}= i

_{X}d + di

_{X}

_{X}ω = i

_{X}dω + di

_{X}ω = di

_{X}ω

_{X}ω = − du = 0

_{X}ω = L

_{Y}ω = 0 and

_{X}L

_{Y}ω − L

_{Y}L

_{X}ω = L

_{[X , Y]}ω = 0

_{h}and locally Hamiltonian one $Z$ one has

_{Z}ω = 0

_{Xh}ω + dh = 0

_{Z}(i

_{Xh}ω + dh) = L

_{[Z , Xh]}ω + i

_{Xh}L

_{Z}ω + dL

_{Z}h

_{[Z , Xh]}ω + dL

_{Z}h = 0

_{h}] is Hamiltonian vector field $X$

_{LZh}, or in other words Hamiltonian vector fields form ideal in algebra of locally Hamiltonian vector fields.

_{ω}can be extended to higher order vector fields and differential forms by linearity and multiplicativity. Namely,

_{ω}(X ∧ Y) = Φ

_{ω}(X) ∧ Φ

_{ω}(Y)

_{ω}is isomorphism, the symplectic form $\omega $ has unique counter image $W$ known as Poisson bivector field. Property $d\omega \; =\; 0$ together with non degeneracy implies that bivector field $W$ is also nondegenerate ($Wn\ne \; 0$) and satisfies condition

_{1}∧ C

_{2}∧ ... ∧ C

_{n}, S

_{1}∧ S

_{2}∧ ... ∧ S

_{n}] =

^{p + q}[C

_{p}, S

_{q}] ∧ C

_{1}∧ C

_{2}∧ ... ∧ Ĉ

_{p}∧ ... ∧ C

_{n}

_{1}∧ S

_{2}∧ ... ∧ Ŝ

_{q}∧ ...∧ S

_{n}

_{W}, transforming differential 1-forms into vector fields, which is inverted to the map $\Phi $

_{ω}and is defined by

_{W}: u → W(u); Φ

_{W}Φ

_{ω}= id

_{f}= W(df)

_{f}, X

_{g}] = X

_{{f , g}}

_{f}∧ X

_{g}) = L

_{Xf}g = − L

_{Xg}g

_{b}bivector field $W$ has the form

_{bc}

_{b}

_{c}

_{b}= W

_{bc}

_{b}

_{z}= e

^{zLE}

_{z}(f) = e

^{zLE}(f) = f + zL

_{E}f + ½(zL

_{E})

^{2}f + ⋯

_{z}(f) = g

_{z}(

_{z}commutes with the vector field $W(h)\; =\; \{h\; ,\; \}$, i. e.

_{z}is called non-Noether symmetry.

^{(k)}=

^{k}∧ W

^{n − k}

^{n}

^{k}∧ W

^{n − k}= Y

^{(k)}W

^{n}.

^{k}∧ W

^{n − k}= (

^{(k)})W

^{n}+ Y

^{(k)}[W(h) , W

^{n}]

^{k − 1}∧ W

^{n − k}+ (n − k)[W(h) , W] ∧ Ŵ

^{k}∧ W

^{n − k − 1}

^{(k)})W

^{n}+ nY

^{(k)}[W(h) , W] ∧ W

^{n − 1}

^{(k)}W

^{n}= 0

_{1}... c

_{n}of the secular equation

^{n}= 0

_{1}... c

_{n}in the following way

^{(k)}=

_{s}> m

_{t}

_{m1}c

_{m2}⋯ c

_{mk}

^{(k)}=

_{E}ω)

^{k}∧ ω

^{n − k}

^{n}

_{1}... c

_{n}can be also derived from the secular equation

_{E}ω − cω)

^{n}= 0

^{n}

^{(k)}= i

_{Wk}(L

_{E}ω)

^{k}

_{E}ω)

^{k}and well defined multivector fields $Wk$. Note that in finite dimensional case the sequence of conservation laws $C(k)$ is related to families of conservation laws $Y(k)$ and $c$

_{k}in the following way

^{(k)}=

_{s}> m

_{t}

_{m1}c

_{m2}⋯ c

_{mk}=

^{(k)}

_{Xh}Y = 0 ⇒

_{E}Y = L

_{Xh}L

_{E}Y = L

_{E}L

_{Xh}Y = 0

_{h}] = 0.

^{(k)}=

^{k}∧ W

^{n − k}

^{n}

_{1}, z

_{2}, z

_{3}, z

_{4}and Poisson bivector field

_{1}

_{3}

_{2}

_{4}

_{1}

^{2}+

_{2}

^{2}+ e

^{z3 − z4}

_{s}

_{s}

_{1}=

_{1}

^{2}− e

^{z3 − z4}−

_{1}+ z

_{2})e

^{z3 − z4}

_{2}=

_{2}

^{2}+ 2e

^{z3 − z4}+

_{1}+ z

_{2})e

^{z3 − z4}

_{3}= 2z

_{1}+

_{2}+

_{1}

^{2}+ e

^{z3 − z4})

_{4}= z

_{2}−

_{1}+

_{2}

^{2}+ e

^{z3 − z4})

_{1}

_{1}

_{3}

_{2}

_{2}

_{4}

^{z3 − z4}

_{1}

_{2}

_{3}

_{4}

_{1}

_{2}

_{3}

_{4}

_{1}+ z

_{2})

_{1}

_{2}

_{3}

_{4}

_{1}z

_{2}− e

^{z3 − z4})

_{1}

_{2}

_{3}

_{4}

^{(1)}=

_{1}+ z

_{2})

^{(2)}=

_{1}z

_{2}− e

^{z3 − z4}

_{1}, z

_{2}, z

_{3}, z

_{4}, z

_{5}, z

_{6}

_{1}

_{4}

_{2}

_{5}

_{3}

_{6}

_{1}

^{2}+

_{2}

^{2}+

_{3}

^{2}+ e

^{z4 − z5}+ e

^{z5 − z6}

_{s}

_{s}

_{1}=

_{1}

^{2}− 2e

^{z4 − z5}−

_{1}+ z

_{2})e

^{z4 − z5}

_{2}=

_{2}

^{2}+ 3e

^{z4 − z5}− e

^{z5 − z6}+

_{1}+ z

_{2})e

^{z4 − z5}

_{3}=

_{3}

^{2}+ 2e

^{z5 − z6}+

_{2}+ z

_{3})e

^{z5 − z6}

_{4}= 3z

_{1}+

_{2}+

_{3}+

_{1}

^{2}+ e

^{z4 − z5})

_{5}= 2z

_{2}−

_{1}+

_{3}+

_{2}

^{2}+ e

^{z4 − z5}+ e

^{z5 − z6})

_{6}= z

_{3}−

_{1}−

_{2}+

_{3}

^{2}+ e

^{z5 − z6})

_{1}

_{1}

_{4}

_{2}

_{2}

_{5}

_{3}

_{3}

_{6}

^{z4 − z5}

_{1}

_{2}

^{z5 − z6}

_{2}

_{3}

_{4}

_{5}

_{5}

_{6}

^{(1)}=

_{1}+ z

_{2}+ z

_{3}) =

^{(2)}=

_{1}z

_{2}+ z

_{1}z

_{3}+ z

_{2}z

_{3}− e

^{z4 − z5}− e

^{z5 − z6}) =

^{(3)}= z

_{1}z

_{2}z

_{3}− z

_{3}e

^{z4 − z5}− z

_{1}e

^{z5 − z6}=

^{n}

_{X}Ω = 0

_{z}(f) = g

_{z}(

_{E}Ω ≠ cΩ

_{E}Ω

_{E}Ω = JΩ.

_{X}L

_{E}Ω = L

_{[X , E]}Ω + L

_{E}L

_{X}Ω

_{X}(JΩ) = (L

_{X}J)Ω + JL

_{X}Ω

_{X}Ω = 0 and vector field $E$ is generator of symmetry satisfying $[E\; ,\; X]\; =\; 0$ commutation relation we obtain

_{X}J)Ω = 0

_{X}J = 0

^{(m)}= (L

_{E})

^{m}Ω

_{4}= z

_{1}

_{5}= z

_{2}

_{6}= z

_{3}

_{1}= − e

^{z4 − z5}

_{2}= e

^{z4 − z5}− e

^{z5 − z6}

_{3}= e

^{z5 − z6}

_{1}∧ dz

_{2}∧ dz

_{3}∧ dz

_{4}∧ dz

_{5}∧ dz

_{6}

_{E}Ω = (z

_{1}+ z

_{2}+ z

_{3}) Ω

_{E}Ω

_{1}+ z

_{2}+ z

_{3}

^{(1)}= L

_{E}J =

_{1}

^{2}+

_{2}

^{2}+

_{3}

^{2}+ e

^{z4 − z5}+ e

^{z5 − z6}

^{(2)}= L

_{E}J

^{(1)}=

_{1}

^{3}+ z

_{2}

^{3}+ z

_{3}

^{3})

_{1}+ z

_{2})e

^{z4 − z5}+

_{2}+ z

_{3})e

^{z5 − z6}

_{P}L

^{(k)}=

^{k})

^{(k)}=

^{k}) =

^{k}) =

^{k − 1}

^{k − 1}[L , P]) =

^{k}, P]) = 0

_{s}, where the bivector field $W$, symplectic form $\omega $ and the generator of the symmetry $E$ have the following form

_{rs}

_{r}

_{r}

_{rs}dz

_{r}∧ dz

_{s}E =

_{s}

_{s}

_{ab}=

_{ad}

_{c}

_{db}

_{c}

_{bc}

_{d}

_{c}

_{dc}

_{b}

_{c}

_{ab}=

_{bc}

_{a}

_{c}

_{bc}

^{2}h

_{a}∂z

_{c}

_{E}(u) = Φ

_{ω}([E , Φ

_{W}(u)]) − L

_{E}u

_{W}and $\Phi $

_{ω}are maps induced by Poisson bivector field and symplectic form). It is remarkable that $\u0154$

_{E}appears to be invariant linear operator. First of all let us show that $\u0154$

_{E}is really linear, or in other words, that for arbitrary 1-forms $u$ and $v$ and function $f$ operator $\u0154$

_{E}has the following properties

_{E}(u + v) = Ŕ

_{E}(u) + Ŕ

_{E}(v)

_{E}(fu) = fŔ

_{E}(u)

_{W}, $\Phi $

_{ω}. Second property can be checked directly

_{E}(fu) = Φ

_{ω}([E , Φ

_{W}(fu)]) − L

_{E}(fu)

_{ω}([E , fΦ

_{W}(u)]) − (L

_{E}f)u − fL

_{E}u

_{ω}((L

_{E}f)Φ

_{W}(u)) + Φ

_{ω}(f[E , Φ

_{W}(u)]) − (L

_{E}f)u − fL

_{E}u

_{E}fΦ

_{ω}Φ

_{W}(u) + fΦ

_{ω}([E , Φ

_{W}(u)]) − (L

_{E}f)u − fL

_{E}u

_{ω}([E , Φ

_{W}(u)]) − L

_{E}u) = fŔ

_{E}(u)

_{ω}Φ

_{W}(u) = u. Now let us check that $\u0154$

_{E}is invariant operator

_{E}= L

_{Xh}Ŕ

_{E}= L

_{Xh}(Φ

_{ω}L

_{E}Φ

_{W}− L

_{E}) = Φ

_{ω}L

_{[Xh , E]}Φ

_{W}− L

_{[Xh, E]}= 0

_{h}commutes with maps $\Phi $

_{W}, $\Phi $

_{ω}(this is consequence of Liouville theorem) and commutes with $E$ as far as $E$ generates the symmetry $[X$

_{h}, E] = 0. In the terms of the local coordinates $\u0154$

_{E}has the following form

_{E}=

_{ab}dz

_{a}⊗

_{b}

_{E}= L

_{W(h)}Ŕ

_{E}= 0

_{E}=

_{ab}dz

_{a}⊗

_{b}

_{ab}) dz

_{a}⊗

_{b}

_{ab}(L

_{W(h)}dz

_{a}) ⊗

_{b}

_{ab}dz

_{a}⊗ (L

_{W(h)}

_{b}

_{ab}) dz

_{a}⊗

_{b}

_{ab}

_{ad}

_{c}

_{d}

_{c}⊗

_{b}

_{ab}W

_{ad}

^{2}h

_{c}∂z

_{d}

_{c}⊗

_{b}

_{ab}

_{cd}

_{b}

_{d}

_{a}⊗

_{c}

_{ab}W

_{cd}

^{2}h

_{b}∂z

_{d}

_{a}⊗

_{c}

_{ab}+

_{ac}L

_{cb}− L

_{ac}P

_{cb})

_{a}⊗

_{b}

_{i}in quite simple way:

^{(k)}=

^{k}) =

_{s}

^{k}

_{E}(89). One can also write down recursion relation that determines conservation laws $I(k)$ in terms of conservation laws $C(k)$

^{(m)}+ (− 1)

^{m}mC

^{(m)}+

^{k}I

^{(m − k)}C

^{(k)}= 0

_{1}^{z3 − z4}_{2}^{z3 − z4}_{1}_{2}^{z3 − z4}^{z3 − z4}^{z3 − z4}^{z3 − z4}^{(1)}=

_{1}+ z

_{2}

^{(2)}=

^{2}) = z

_{1}

^{2}+ z

_{2}

^{2}+ 2e

^{z3 − z4}

_{1}^{z4 − z5}_{2}^{z4 − z5}^{z5 − z6}_{3}^{z5 − z6}_{1}_{2}_{3}^{z4 − z5}^{z4 − z5}^{z4 − z5}^{z4 − z5}− e^{z5 − z6}^{z5 − z6}^{z5 − z6}^{z5 − z6}^{(1)}=

_{1}+ z

_{2}

^{(2)}=

^{2}) = z

_{1}

^{2}+ z

_{2}

^{2}+ z

_{3}

^{2}+ 2e

^{z4 − z5}+ 2e

^{z5 − z6}

^{(3)}=

^{3}) = z

_{1}

^{3}+ z

_{2}

^{3}+ z

_{3}

^{3}+ 3(z

_{1}+ z

_{2})e

^{z4 − z5}+ 3(z

_{2}+ z

_{3})e

^{z5 − z6}

_{1}, c

_{2}... c

_{n}as linear independence of either differentials of conservation laws $dc$

_{1}, dc

_{2}... dc

_{n}or corresponding Hamiltonian vector fields $X$

_{c1}, X

_{c2}... X

_{cn}. Strictly speaking we can say that conservation laws $c$

_{1}, c

_{2}... c

_{n}are functionally independent if Lesbegue measure of the set of points of phase space $M$ where differentials $dc$

_{1}, dc

_{2}... dc

_{n}become linearly dependent is zero. Involutivity of conservation laws means that all possible Poisson brackets of these conservation laws vanish pair wise

_{i}, c

_{j}} = 0 i, j = 1... n

_{1}, c

_{2}... c

_{n}implies that corresponding Hamiltonian vector fields $X$

_{c1}, X

_{c2}... X

_{cn}span Lagrangian subspace (isotropic subspace of dimension $n$) of tangent space (at each point of $M$). Indeed, due to property (23)

_{i}, c

_{j}} = ω(X

_{ci}, X

_{cj}) = 0

_{c1}, X

_{c2}... X

_{cn}is isotropic. Dimension of this space is $n$ so it is Lagrangian. Note also that distribution $X$

_{c1}, X

_{c2}... X

_{cn}is integrable since due to (22)

_{ci}, X

_{cj}] = X

_{{ci , cj}}= 0

_{c1}, X

_{c2}... X

_{cn}spans tangent space of this submanifold. Thus for phase space geometry existence of complete involutive set of integrals of motion implies existence of invariant Lagrangian submanifold.

^{(k)}, Y

^{(m)}} = 0

_{W(f)}W = [W(f) , W] = 0

_{E}[W , W] = [E[W , W]] = [[E , W] W] + [W[E , W]] = 2[Ŵ , W] = 0.

_{i}≠ c

_{j}of the equation (40). By taking the Lie derivative of the equation

_{i}W)

^{n}= 0

_{j}) and $\u0174(c$

_{j}) and using Liouville theorem for $W$ and $\u0174$ bivectors we obtain the following relations

_{i}W)

^{n − 1}(L

_{W(cj)}Ŵ − {c

_{j}, c

_{i}}W) = 0,

_{i}W)

^{n − 1}(c

_{i}L

_{Ŵ(cj)}W + {c

_{j}, c

_{i}}

_{∗}W) = 0,

_{i}, c

_{j}}

_{∗}= Ŵ(dc

_{i}∧ dc

_{j})

_{i}subtracting (120) and using identity (114) gives rise to

_{i}, c

_{j}}

_{∗}− c

_{i}{c

_{i}, c

_{j}})(Ŵ − c

_{i}W)

^{n − 1}W = 0

_{i}, c

_{j}}

_{∗}− c

_{i}{c

_{i}, c

_{j}} = 0

_{i}W)

^{n − 1}W vanishes. In the second case we can repeat (119)-(122) procedure for the volume field $(\u0174\; -\; c$

_{i}W)

^{n − 1}W yielding after $n$ iterations $Wn=\; 0$ that according to our assumption (that the dynamical system is regular) is not true. As a result we arrived at (123) and by the simple interchange of indices $i\; \leftrightarrow \; j$ we get

_{i}, c

_{j}}

_{∗}− c

_{j}{c

_{i}, c

_{j}} = 0

_{i}are in involution with respect to the both Poisson structures (since $c$

_{i}≠ c

_{j})

_{i}, c

_{j}}

_{∗}= {c

_{i}, c

_{j}} = 0

^{n}≠ 0

_{1}

_{3}

_{2}

_{4}

_{1}

_{1}

_{3}

_{2}

_{2}

_{4}

^{z3 − z4}

_{1}

_{2}

_{3}

_{4}

_{1}

_{4}

_{2}

_{5}

_{3}

_{6}

_{1}

_{1}

_{4}

_{2}

_{2}

_{5}

_{3}

_{3}

_{6}

^{z4 − z5}

_{1}

_{2}

^{z5 − z6}

_{2}

_{3}

_{4}

_{5}

_{5}

_{6}

_{E}ω (clearly $d\omega \ast =\; dL$

_{E}ω = L

_{E}dω = 0). It is important that by taking Lie derivative of Hamilton's equation

_{Xh}ω + dh = 0

_{E}(i

_{Xh}ω + dh) = i

_{[E , Xh]}ω + i

_{Xh}L

_{E}ω + L

_{E}dh = i

_{Xh}ω

^{∗}+ dL

_{E}h = 0

_{Xh}ω

^{∗}+ dh

^{∗}= 0

_{E}h. This is actually second Hamiltonian realization of equations of motion and thus under certain conditions existence of non-Noether symmetry gives rise to additional presymplectic structure $\omega \ast $ 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 and Benney system) non-Noether symmetries that are responsible for existence of bi-Hamiltonian structures has been found and motivated further investigation of relationship between symmetries and bi-Hamiltonian structures. Namely it seems to be interesting to know whether 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-forms $\omega $ and $\omega \ast $ such that

^{∗}= 0, ω

^{n}≠ 0

_{Xh}ω + dh = 0

_{Xh}ω

^{∗}+ dh

^{∗}= 0

_{h}] = 0 and $\omega \ast =\; L$

_{E}ω.

_{E∗}ω = θ

^{∗}

_{E∗}ω = ω

^{∗}

_{E∗}ω = di

_{E∗}ω + i

_{E∗}dω = dθ

^{∗}= ω

^{∗}

_{[E∗, Xh]}ω = L

_{E∗}(i

_{Xh}ω) − i

_{Xh}L

_{E∗}ω = − d(E

^{∗}(h) − h

^{∗}) = − dh'

_{h}, E

^{∗}] is Hamiltonian vector field

_{h}, E] = X

_{h'}

_{g}, that satisfies the same commutation relation. Namely let us define function (in general case this can be done only locally)

_{h}, X

_{g}] = X

_{h'}). Using $E\ast $ and $X$

_{h'}one can construct generator of non-Noether symmetry — non-Hamiltonian vector field $E\; =\; E\ast -\; X$

_{g}commuting with $X$

_{h}and satisfying

_{E}ω = L

_{E∗}ω − L

_{Xg}ω = L

_{E∗}ω = ω

^{∗}

_{Xg}ω = 0). So in case of regular Hamiltonian system every exact bi-Hamiltonian structure is naturally associated with some (non-Noether) symmetry of space of solutions. In case where bi-Hamiltonian structure is not exact ($\omega \ast $ is closed but not exact) then due to

^{∗}= L

_{E}ω = di

_{E}ω + i

_{E}dω = di

_{E}ω

^{(k)}

^{(k)}→ Ω

^{(k + 1)}

_{ω}([W , Φ

_{W}(u)])

_{ω}([[E , W]Φ

_{W}(u)])

_{W}, Φ

_{ω}maps. Then, if $u$ is a $k$-degree form $\Phi $

_{W}maps it on $k$-degree multivector field and the Schouten brackets $[W\; ,\; \Phi $

_{W}(u)] and $[[E\; ,\; W]\Phi $

_{W}(u)] result the $k\; +\; 1$-degree multivector fields that are mapped on $k\; +\; 1$-degree differential forms by $\Phi $

_{ω}. So, $d$ and $\u0111$ are linear maps from $\Omega (k)$ into $\Omega (k\; +\; 1)$. Derivation property follows from the same feature of the Schouten bracket $[\; ,\; ]$ and linearity of $\Phi $

_{W}and $\Phi $

_{ω}maps. Now we have to prove the nilpotency of $d$ and $\u0111$. Let us consider $d2u$

^{2}u = Φ

_{ω}([W , Φ

_{W}(Φ

_{ω}([W , Φ

_{W}(u)]))]) = Φ

_{ω}([W[W , Φ

_{W}(u)]]) = 0

^{2}u = Φ

_{ω}([[W , E][[W , E]Φ

_{W}(u)]]) = 0

_{ω}([[[W , E]W]Φ

_{W}(u)]) = 0

^{(k)}= kdI

^{(k + 1)}

_{1}= z

_{1}dz

_{1}− e

^{z3 − z4}dz

_{4}

_{2}= z

_{2}dz

_{2}+ e

^{z3 − z4}dz

_{3}

_{3}= z

_{1}dz

_{3}+ dz

_{2}

_{4}= z

_{2}dz

_{4}− dz

_{1}

_{1}= 0. Indeed

^{2}z

_{1}= đđz

_{1}= đ(z

_{1}dz

_{1}− e

^{z3 − z4}dz

_{4})

_{1}∧ dz

_{1}+ z

_{1}đdz

_{1}− e

^{z3 − z4}đz

_{3}∧ dz

_{4}+ e

^{z3 − z4}đz

_{4}∧ dz

_{4}− e

^{z3 − z4}đdz

_{4}

_{1}∧ dz

_{1}− z

_{1}dđz

_{1}− e

^{z3 − z4}đz

_{3}∧ dz

_{4}+ e

^{z3 − z4}đz

_{4}∧ dz

_{4}+ e

^{z3 − z4}dđz

_{4}= 0

_{1}∧ dz

_{1}= e

^{z3 − z4}dz

_{1}∧ dz

_{4},

_{1}dđz

_{1}= z

_{1}e

^{z3 − z4}dz

_{3}∧ dz

_{4},

^{z3 − z4}đz

_{3}∧ dz

_{4}

_{1}e

^{z3 − z4}dz

_{1}∧ dz

_{4}− e

^{z3 − z4}dz

_{2}∧ dz

_{4},

^{z3 − z4}đz

_{4}∧ dz

_{4}= e

^{z3 − z4}dz

_{2}∧ dz

_{4}

^{z3 − z4}dđz

_{4}= − e

^{z3 − z4}dz

_{1}∧ dz

_{4}

^{2}z

_{2}= đ

^{2}z

_{3}= đ

^{2}z

_{4}= 0

^{(1)}= z

_{1}+ z

_{2}

^{(2)}= z

_{1}

^{2}+ z

_{2}

^{2}+ 2e

^{z3 − z4}

^{(1)}= dI

^{(2)}

_{1}= z

_{1}dz

_{1}− e

^{z4 − z5}dz

_{5}

_{2}= z

_{2}dz

_{2}+ e

^{z4 − z5}dz

_{4}− e

^{z5 − z6}dz

_{6}

_{3}= z

_{3}dz

_{3}+ e

^{z5 − z6}dz

_{5}

_{4}= z

_{1}dz

_{4}− dz

_{2}− dz

_{3}

_{5}= z

_{2}dz

_{5}+ dz

_{1}− dz

_{3}

_{6}= z

_{3}dz

_{6}+ dz

_{1}+ dz

_{2}

^{(1)}= z

_{1}+ z

_{2}

^{(2)}= z

_{1}

^{2}+ z

_{2}

^{2}+ z

_{3}

^{2}+ 2e

^{z4 − z5}+ 2e

^{z5 − z6}

^{(3)}= z

_{1}

^{3}+ z

_{2}

^{3}+ z

_{3}

^{3}+ 3(z

_{1}+ z

_{2})e

^{z4 − z5}+ 3(z

_{2}+ z

_{3})e

^{z5 − z6}

^{(1)}= dI

^{(2)}

^{(2)}= 2dI

^{(3)}

_{E}(89) and to show how Frölicher-Nijenhuis geometry arises in Hamiltonian system that possesses certain non-Noether symmetry. From the geometric properties of the tangent valued forms we know that the traces of powers of a linear operator $F$ on tangent bundle are in involution whenever its Frölicher-Nijenhuis torsion $T(F)$ vanishes, i. e. whenever for arbitrary vector fields $X,Y$ the condition

_{E}(X) = Φ

_{W}(L

_{E}Φ

_{ω}(X)) − [E , X]

_{E}follows from the invariance of the $\u0154$

_{E}defined by (89) (note that for arbitrary 1-form vector field $u$ and vector field $X$ contraction $i$

_{X}u has the property $i$

_{REX}u = i

_{X}Ŕ

_{E}u, so $R$

_{E}is actually transposed to $\u0154$

_{E}). It remains to show that the condition (110) ensures vanishing of the Frölicher-Nijenhuis torsion $T(R$

_{E}) of $R$

_{E}, i.e. for arbitrary vector fields $X,\; Y$ we must get

_{E})(X , Y) = [R

_{E}(X) , R

_{E}(Y)] − R

_{E}([R

_{E}(X) , Y]

_{E}(Y)] − R

_{E}([X , Y])) = 0

_{ω}(W), ω

^{∗}= Ŕ

_{E}ω ω

^{∗∗}= Ŕ

_{E}ω

^{∗}

_{ω}([W , W]) = 0

^{∗}= dΦ

_{ω}([E , W]) − dL

_{E}ω = Φ

_{ω}([[E , W]W]) − L

_{E}dω = 0

_{ω}([E , W]) and using the condition (110), we get

^{∗∗}= 2Φ

_{ω}([[E[E , W]]W]) − 2dL

_{E}ω

^{∗}= − 2L

_{E}dω

^{∗}= 0

^{∗}= dω

^{∗∗}= 0

_{E}) and $\omega $.

_{T(RE)(X , Y)}ω = i

_{[REX , REY]}ω − i

_{[REX , Y]}ω

^{∗}− i

_{[X , REY]}ω

^{∗}+ i

_{[X , Y]}ω

^{∗∗}

_{REX}i

_{Y}ω

^{∗}− i

_{REY}L

_{X}ω

^{∗}− L

_{REX}i

_{Y}ω

^{∗}+ i

_{Y}L

_{REX}ω

^{∗}− L

_{X}i

_{REY}ω

^{∗}+ i

_{REY}L

_{X}ω

^{∗}+ i

_{[X , Y]}ω

^{∗∗}

_{Y}L

_{X}ω

^{∗∗}− L

_{X}i

_{Y}ω

^{∗∗}+ i

_{[X , Y]}ω

^{∗∗}= 0

_{X}i

_{Y}ω = i

_{Y}L

_{X}ω + i

_{[X , Y]}ω

_{REX}ω = di

_{REX}ω + i

_{REX}dω = di

_{X}ω

^{∗}

_{X}ω

^{∗}− i

_{X}dω

^{∗}= L

_{X}ω

^{∗}

_{E})(X , Y) and $\omega $ vanishes. But since $W$ bivector is non-degenerate ($Wn\ne \; 0$), its counter image

_{ω}(W)

_{E}) itself is zero. So we get

_{E})(X , Y) = [R

_{E}(X) , R

_{E}(Y)] − R

_{E}([R

_{E}(X) , Y]

_{E}(Y)] − R

_{E}([X , Y])) = 0

_{E}associated with non-Noether symmetry (53) reproduces well known Frölicher-Nijenhuis operator

_{E}= z

_{1}dz

_{1}⊗

_{1}

_{1}⊗

_{4}

_{2}dz

_{2}⊗

_{2}

_{2}⊗

_{3}

_{1}dz

_{3}⊗

_{3}

^{z3 − z4}dz

_{3}⊗

_{2}

_{2}dz

_{4}⊗

_{4}

^{z3 − z4}dz

_{4}⊗

_{1}

_{E}plays the role of recursion operator for conservation laws

^{(1)}= z

_{1}+ z

_{2}

^{(2)}= z

_{1}

^{2}+ z

_{2}

^{2}+ 2e

^{z3 − z4}

_{E}(dI

^{(1)}) = dI

^{(2)}

_{E}= z

_{1}dz

_{1}⊗

_{1}

^{z4 − z5}dz

_{5}⊗

_{1}

_{2}dz

_{2}⊗

_{2}

^{z4 − z5}dz

_{4}⊗

_{2}

^{z5 − z6}dz

_{6}⊗

_{2}

_{3}dz

_{3}⊗

_{3}

^{z5 − z6}dz

_{5}⊗

_{3}

_{1}dz

_{4}⊗

_{4}

_{2}⊗

_{4}

_{3}⊗

_{4}

_{2}dz

_{5}⊗

_{5}

_{1}⊗

_{5}

_{3}⊗

_{5}

_{3}dz

_{6}⊗

_{6}

_{1}⊗

_{6}

_{2}⊗

_{6}

_{E}appears to be recursion operator for conservation laws

^{(1)}= z

_{1}+ z

_{2}

^{(2)}= z

_{1}

^{2}+ z

_{2}

^{2}+ z

_{3}

^{2}+ 2e

^{z4 − z5}+ 2e

^{z5 − z6}

^{(3)}= z

_{1}

^{3}+ z

_{2}

^{3}+ z

_{3}

^{3}

_{1}+ z

_{2})e

^{z4 − z5}+ 3(z

_{2}+ z

_{3})e

^{z5 − z6}

^{(3)}= 3Ŕ

_{E}(dI

^{(2)}) = 6(Ŕ

_{E})

^{2}(dI

^{(1)})

_{z}defined by (28) naturally acts on algebra of integrals of motion. Namely for each conservation law

_{z}to $J$

_{z}(J) = e

^{zLE}J = J + zL

_{E}J + ½(zL

_{E})

^{2}J + ...

_{z}(J) = g

_{z}(

_{a}.

_{1}) , J(z

_{2})} = 0

_{1}, z

_{2}). On $2n$ dimensional symplectic manifold each involutive family that contains $n$ functionally independent integrals 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 it is 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 of generator $E$ of group $g$

_{z}= e

^{zLE}. In other words we would like to know what condition must be satisfied by generator of symmetry $E$ and integral of motion $J$ to ensure that $\{J(z$

_{1}) , J(z

_{2})} = 0. To address this issue and to describe class of vector fields that possess nontrivial involutive orbits we would like to propose the following theorem

_{0}[W(s)[W(s) ,W]] (c

_{0}≠ − 1)

_{W(s)}dJ) = c

_{1}[W(s),W](dJ) (c

_{1}≠ 0)

_{0,1}are some constants) gives rise to involutive set of functions

^{(m)}= (L

_{W(s)})

^{m}J {J

^{(m)}, J

^{(k)}} = 0

_{W}(L

_{W(s)}Φ

_{ω}(V)))

^{2}(W) = R([W(s),W]) = ½([W(s)[W(s),W]] − Φ

_{W}((L

_{W(s)})

^{2}ω))

_{0})[W(s)[W(s),W]]

_{W}((L

_{W(s)})

^{2}ω) = Φ

_{W}(L

_{W(s)}L

_{W(s)}ω)

_{W}(i

_{W(s)}dL

_{W(s)}ω) + Φ

_{W}(di

_{W(s)}L

_{W(s)}ω)

_{W}(i

_{W(s)}L

_{W(s)}ω)] = [W[W(s),W](s)] = c

_{0}[W(s)[W(s),W]]

_{W(s)}R + R

^{2})(W)

_{0})(L

_{W(s)}R + R

^{2}) = 2R

^{2}

_{0})L

_{W(s)}R = (1 − c

_{0})R

^{2}

_{W(s)}dJ) = c

_{1}R(W)(dJ)

^{m}(W)(L

_{W(s)}dJ) = c

_{1}R

^{m + 1}(W)(dJ)

_{W(s)})

^{m}dJ) = c

_{m}R

^{m}(W)(dJ)

_{W(s)})

^{m}dJ) + W((L

_{W(s)})

^{m + 1}dJ)

_{m}

_{0}

_{0}

^{m + 1}(W)(dJ) + c

_{m}R

^{m + 1}(W)(dJ)

^{k}W((L

_{W(s)})

^{m}dJ) = c

_{m}R

^{k + m}(W)(dJ)

_{W(s)})

^{m + 1}dJ) = c

_{m + 1}R

^{m + 1}(W)(dJ)

_{m + 1}is defined by

_{0})c

_{m + 1}= mc

_{n}(1 − c

_{0})

_{n}can be determined by

_{m}(1 + c

_{0})

^{m − 1}= c

_{0}(m − 1)!(1 − c

_{0})

^{m − 1}

_{W(s)})

^{m}J are in involution. Indeed

_{W(s)})

^{m}J, (L

_{W(s)})

^{k}J} = W(d(L

_{W(s)})

^{m}J ∧ d(L

_{W(s)})

^{k}J)

_{W(s)})

^{m}dJ ∧ (L

_{W(s)})

^{k}dJ) = c

_{m}c

_{k}W(dJ ∧ dJ) = 0

_{s}= p

_{s}

_{s}= ε(s − 1)e

^{qs − 1 − qs}− ε(n − s)e

^{qs − qs + 1}

_{s}

_{s}

_{s}

^{2}+

^{qs − qs + 1}

_{s}= p

_{s}

_{n + s}= q

_{s}s = 1, 2, (3); n = 2(3)

_{z}generated by the vector field $E$ will be symmetry of Toda chain if for each $p$

_{s}, q

_{s}satisfying Toda equations (214) $g$

_{z}(p

_{s}), g

_{z}(q

_{s}) also satisfy it. Substituting infinitesimal transformations

_{z}(p

_{s}) = p

_{s}+ zE(p

_{s}) + O(z

^{2})

_{z}(p

_{s}) = q

_{s}+ zE(q

_{s}) + O(z

^{2})

_{s}) = E(p

_{s})

_{s}) = ε(s − 1)e

^{qs − 1 − qs}(E(q

_{s − 1}) − E(q

_{s})) − ε(n − s)e

^{qs − qs + 1}(E(q

_{s}) − E(q

_{s + 1}))

_{s}) = ½p

_{s}

^{2}+ ε(s − 1)(n − s + 2)e

^{qs − 1 − qs}− ε(n − s)(n − s) e

^{qs − qs + 1}

_{s − 1}+ p

_{s}) e

^{qs − 1 − qs}− ε(n − s)(p

_{s}+ p

_{s + 1}) e

^{qs − qs + 1})

_{s}) = (n − s + 1)p

_{s}− ½

_{k}+ ½

_{k}

_{s}

^{2}+ ε(s − 1)e

^{qs − 1 − qs}+ ε(n − s)e

^{qs − qs + 1})

_{E}W leads to the following Poisson bivector field

_{s}

_{s}

_{s}

^{qs − qs + 1}

_{s}

_{s + 1}

_{s}

_{r}

_{E}W give rise to bi-Hamiltonian structure of Toda model (compare with [30]). Thus bi-Hamiltonian realization of Toda chain can be considered as manifestation of hidden symmetry. In terms of bivector fields these bi-Hamiltonian system is formed by The conservation laws (45) associated with the symmetry reproduce well known set of conservation laws of Toda chain.

^{(1)}= C

^{(1)}=

_{s}

^{(2)}= (C

^{(1)})

^{2}− 2C

^{(2)}=

_{s}

^{2}+ 2

^{qs − qs + 1}

^{(3)}= C

^{(1)})

^{3}− 3C

^{(1)}C

^{(2)}+ 3C

^{(3)}=

_{s}

^{3}+ 3

_{s}+ p

_{s + 1}) e

^{qs − qs + 1}

^{(4)}= C

^{(1)})

^{4}− 4(C

^{(1)})

^{2}C

^{(2)}+ 2(C

^{(2)})

^{2}+ 4C

^{(1)}C

^{(3)}− 4C

^{(4)}

_{s}

^{4}+ 4

_{s}

^{2}+ 2p

_{s}p

_{s + 1}+ p

_{s + 1}

^{2}) e

^{qs − qs + 1}

^{2(qs − qs + 1)}+ 4

^{qs − qs + 2}

^{(m)}= (− 1)

^{m + 1}mC

^{(m)}+

^{k + 1}I

^{(m − k)}C

^{(k)}

_{k, k}= L

_{n + k, n + k}= p

_{k}

_{n + k, k + 1}= − L

_{n + k + 1, k}= ε(n − k)e

^{qk − qk + 1}

_{k, n + m}= ε(m − k)

_{k, n + k}= 1

_{n + k, k}= − ε(k − 1)e

^{qk − 1 − qk}− ε(n − k)e

^{qk − qk + 1}

_{n + k, k + 1}= ε(n − k)e

^{qk − qk + 1}

_{n + k, k − 1}= ε(k − 1)e

^{qk − 1 − qk}

_{s}= p

_{s}dq

_{s}+

_{r}−

_{r}

_{s}= p

_{s}dp

_{s}− e

^{qs − qs + 1}dq

_{s + 1}+ e

^{qs − 1 − qs}dq

_{s}

^{(k)}= kdI

^{(k + 1)}

_{E}=

_{s}(dp

_{s}⊗

_{s}

_{s}⊗

_{s}

^{qs − qs + 1}dq

_{s + 1}⊗

_{s}

^{qs − 1 − qs}dq

_{s}⊗

_{s}

_{s}⊗

_{r}

_{r}⊗

_{s}

_{E}(dI

^{(k)}) = kdI

^{(k + 1)}

^{(1)}+

^{2}

^{(2)}+

^{3}

^{(3)}+ ⋯

^{(m)}= (L

_{E})

^{m}J

_{a}= e

^{aLE}on total momenta of Toda chain

_{s}

^{(1)}= L

_{E}J = ½

_{s}

^{2}+

^{qs − qs + 1}

^{(2)}= L

_{E}J

^{(1)}= (L

_{E})

^{2}J =

_{s}

^{3}+

_{s}+ p

_{s + 1}) e

^{qs − qs + 1}

^{(3)}= L

_{E}J

^{(2)}= (L

_{E})

^{3}J = ¾

_{s}

^{4}+ 3

_{s}

^{2}+ 2p

_{s}p

_{s + 1}+ p

_{s + 1}

^{2})e

^{qs − qs + 1}

^{2(qs − qs + 1)}+ 3

^{qs − qs + 2}

^{(m)}= L

_{E}J

^{(m − 1)}= (L

_{E})

^{m}J

_{W(s)}dJ) = − [W(s),W](dJ)

_{t}+ u

_{xxx}+ uu

_{x}= 0

_{t}+ E(u)

_{xxx}+ u

_{x}E(u) + uE(u)

_{x}= 0

_{xx}+

^{2}+

_{x}v +

_{xxx}+ uu

_{x}) −

_{xxxxx}+ 20u

_{x}u

_{xx}+ 10 uu

_{xxx}+ 5u

^{2}u

_{x})

_{x}= u).

_{t}= {h , u}

_{x}

_{x}

^{2}−

^{3}

_{xx}∧ {

_{x}+

_{x})

^{(0)}=

_{E})

^{m}J

^{(0)}that reproduce well known conservation laws of KdV equation

^{(0)}=

^{(1)}= L

_{E}J

^{(0)}= ¼

^{2}dx

^{(2)}= (L

_{E})

^{2}J

^{(0)}=

^{3}

_{x}

^{2}) dx

^{(3)}= (L

_{E})

^{3}J

^{(0)}

^{4}−

_{x}

^{2}+ u

_{xx}

^{2}) dx

^{(m)}= (L

_{E})

^{m}J

^{(0)}

_{t}= u

_{x}w + uw

_{x}

_{t}= uu

_{x}− v

_{xx}+ 2v

_{x}w + 2vw

_{x}

_{t}= w

_{xx}− 2v

_{x}+ 2ww

_{x}

_{t}= (wE(u))

_{x}+ (uE(w))

_{x}

_{t}= (uE(u))

_{x}− E(v)

_{xx}+ 2(wE(v))

_{x}+ 2(vE(w))

_{x}

_{t}= E(w)

_{xx}− 2E(v)

_{x}+ 2(wE(w))

_{x}

^{2})

^{2})

^{2})

_{x}+ 2t(uw

^{2}− 2uv + uw

_{x})

_{x}

^{2}+ 4vw − 3v

_{x}+ x(uu

_{x}+ 2(vw)

_{x}− v

_{xx})

^{2}w − uu

_{x}− 3v

^{2}+ 3vw

^{2}− 3v

_{x}w + v

_{xx})

_{x}

^{2}+ 2w

_{x}− 4v + x(2ww

_{x}+ w

_{xx}− 2v

_{x})

^{2}+ 6vw − w

^{3}− 3ww

_{x}− w

_{xx})

_{x}

_{x}= u).

_{t}= {h , u}

_{t}= {h , v}

_{t}= {h , w}

^{2}w + 2vw

^{2}− 2v

_{x}w − 2v

^{2})dx

_{x}+

_{x}} dx

_{x}+ v

_{x}

_{x}∧ {

_{x}+ w

_{x}+ {

_{x}∧

_{t}= {h

^{∗}, u}

_{∗}

_{t}= {h

^{∗}, v}

_{∗}

_{t}= {h

^{∗}, w}

_{∗}

^{∗}= − ¼

^{2}+ 2vw)dx

_{∗}is Poisson bracket defined by bivector field $\u0174$.

^{(0)}=

^{(0)}=

^{(1)}= L

_{E}J

^{(0)}= − 2

^{(2)}= L

_{E}J

^{(1)}= (L

_{E})

^{2}J

^{(0)}= − 2

^{2}+ 2vw)dx

^{(3)}= L

_{E}J

^{(2)}= (L

_{E})

^{3}J

^{(0)}= − 6

^{2}w + 2vw

^{2}− 2v

_{x}w − 2v

^{2})dx

^{(4)}= L

_{E}J

^{(3)}= (L

_{E})

^{4}J

^{(0)}

^{2}w

^{2}+ u

^{2}w

_{x}− 2u

^{2}v − 6v

^{2}w + 2vw

^{3}− 3v

_{x}w

^{2}− 2v

_{x}w

_{x})dx

^{(n)}= L

_{E}J

^{(n − 1)}= (L

_{E})

^{n}J

^{(0)}

_{t}= u

_{x}w + uw

_{x}

_{t}= uu

_{x}+ 2v

_{x}w + 2vw

_{x}

_{t}= − 2v

_{x}+ 2ww

_{x}

_{x}+ 2t(uw

^{2}− 2uv)

_{x}

^{2}+ 4vw + x(uu

_{x}+ 2(vw)

_{x}) + 2t(u

^{2}w − 3v

^{2}+ 3vw

^{2})

_{x}

^{2}− 4v + x(2ww

_{x}− 2v

_{x}) − 2t(u

^{2}+ 6vw − w

^{3})

_{x}

^{(0)}=

^{(1)}= L

_{E}J

^{(0)}= − 2

^{(2)}= L

_{E}J

^{(1)}= (L

_{E})

^{2}J

^{(0)}= − 2

^{2}+ 2vw)dx

^{(3)}= L

_{E}J

^{(2)}= (L

_{E})

^{3}J

^{(0)}= − 6

^{2}w + 2vw

^{2}− 2v

^{2})dx

^{(4)}= L

_{E}J

^{(3)}= (L

_{E})

^{4}J

^{(0)}= − 24

^{2}w

^{2}− 2u

^{2}v − 6v

^{2}w + 2vw

^{3})dx

^{(n)}= L

_{E}J

^{(n − 1)}= (L

_{E})

^{n}J

^{(0)}

_{t}= ½ v

_{xx}+ v

_{x}w + vw

_{x}

_{t}= − ½ w

_{xx}+ v

_{x}+ ww

_{x}

_{x}+ x(2(vw)

_{x}+ v

_{xx})

^{2}+ 3vw

^{2}+ 3v

_{x}w + v

_{xx})

_{x}

^{2}− 2w

_{x}+ 4v + x(2ww

_{x}− w

_{xx}+ 2v

_{x})

^{3}− 3ww

_{x}+ w

_{xx})

_{x}

^{(0)}=

^{(1)}= L

_{E}J

^{(0)}= 2

^{(2)}= L

_{E}J

^{(1)}= (L

_{E})

^{2}J

^{(0)}= 4

^{(3)}= L

_{E}J

^{(2)}= (L

_{E})

^{3}J

^{(0)}= 12

^{2}+ v

_{x}w + v

^{2})dx

^{(4)}= L

_{E}J

^{(3)}= (L

_{E})

^{4}J

^{(0)}= 24

^{2}w + 2vw

^{3}+ 3v

_{x}w

^{2}− 2v

_{x}w

_{x})dx

^{(n)}= L

_{E}J

^{(n − 1)}= (L

_{E})

^{n}J

^{(0)}

_{t}= {h , v}

_{t}= {h , w}

^{2}+ v

_{x}w + v

^{2})dx

_{x}} dx

_{x}

_{x}∧ {

_{x}+ w

_{x}+

_{x}} dx

_{t}= {h

^{∗}, v}

_{∗}

_{t}= {h

^{∗}, w}

_{∗}

^{∗}= − ¼

_{t}= v

_{x}w + vw

_{x}

_{t}= v

_{x}+ ww

_{x}

_{x}+ 3t(v

^{2}+ vw

^{2})

_{x}

^{2}+ 4v + 2x(ww

_{x}+ v

_{x}) + t(6vw + w

^{3})

_{x}

^{(0)}=

^{(1)}= L

_{E}J

^{(0)}= 2

^{(2)}= L

_{E}J

^{(1)}= (L

_{E})

^{2}J

^{(0)}= 4

^{(3)}= L

_{E}J

^{(2)}= (L

_{E})

^{3}J

^{(0)}= 12

^{2}+ v

^{2})dx

^{(4)}= L

_{E}J

^{(3)}= (L

_{E})

^{4}J

^{(0)}= 48

^{2}w + vw

^{3})dx

^{(n)}= L

_{E}J

^{(n − 1)}= (L

_{E})

^{n}J

^{(0)}

_{x}} dx

_{x}

_{x}+

_{x}} dx

_{t}= w

_{xx}+ ww

_{x}

_{t}= vv

_{x}+ 2(uw)

_{x}

_{t}= 2u

_{x}+ (vw)

_{x}

_{t}= 2v

_{x}+ 2ww

_{x}

_{t}= (vE(v))

_{x}+ 2(uE(w))

_{x}+ 2(wE(u))

_{x}

_{t}= 2E(u)

_{x}+ (vE(w))

_{x}+ (wE(v))

_{x}

_{t}= 2E(v)

_{x}+ 2(wE(w))

_{x}

^{2})

^{2})

^{2})

^{2}+ x(2(uw)

_{x}+ vv

_{x}) + 2t(4uv + v

^{2}w + 3uw

^{2})

_{x}

_{x}+ 2u

_{x}) + 2t(4uw + 3v

^{2}+ vw

^{2})

_{x}

^{2}+ 4v + 2x(ww

_{x}+ v

_{x}) + 2t(w

^{3}+ 4vw + 4u)

_{x}

_{t}= {h , u}

_{t}= {h , v}

_{t}= {h , w}

^{2}+ 4uv + v

^{2}w)dx

_{x}+

_{x}} dx

_{x}+ v

_{x}

_{x}+ 2

_{x}} dx

_{t}= {h

^{∗}, u}

_{∗}

_{t}= {h

^{∗}, v}

_{∗}

_{t}= {h

^{∗}, w}

_{∗}

^{∗}=

^{2}+ 2uw)dx

^{(0)}=

^{(0)}=

^{(1)}= L

_{E}J

^{(0)}= 2

^{(2)}= L

_{E}J

^{(1)}= (L

_{E})

^{2}J

^{(0)}= 8

^{(3)}= L

_{E}J

^{(2)}= (L

_{E})

^{3}J

^{(0)}= 12

^{2}+ 2uw)dx

^{(4)}= L

_{E}J

^{(3)}= (L

_{E})

^{4}J

^{(0)}= 48

^{2}+ 4uv + v

^{2}w)dx

^{(5)}= L

_{E}J

^{(4)}= (L

_{E})

^{5}J

^{(0)}= 240

^{2}+ 8uvw + 2uw

^{3}+ 2v

^{3}+ v

^{2}w

^{2})dx

^{(n)}= L

_{E}J

^{(n − 1)}= (L

_{E})

^{n}J

^{(0)}