In Hamiltonian integrable models, conservation laws often form involutive orbit of
one-parameter symmetry group. Such a symmetry carries important information about
integrable model and its bi-Hamiltonian structure. The present paper is an attempt to
describe class of one-parameter group of transformations of Poisson manifold
that possess involutive orbits and may be related to Hamiltonian integrable systems.

Let *C*^{∞}(M) be algebra of smooth functions on manifold *M* equipped with Poisson bracket
*W* is Poisson bivector satisfying property *[W , W] = 0*.
Each vector field *E* on manifold *M* gives rise to one-parameter group of transformations of
*C*^{∞}(M) algebra
*L*_{E} denotes Lie derivative along the vector field *E*.
To any smooth function *J ∈ C*^{∞}(M) this group assigns orbit that goes through *J*
*J(z)* is called involutive if
*J(z)*
plays the role of involutive family of conservation laws.

{f , g} = W(df ∧ dg)

where
g_{z} = e^{zLE}

where
J(z) = g_{z}(J) = e^{zLE}(J) = J + zL_{E}J + ½z²(L_{E})²J + ⋯

the orbit
{J(x) , J(y)} = 0 ∀x, y ∈ ℝ

Involutive orbits are often related to integrable models where
Involutivity of orbit *J(z)* depends on nature of vector field *E* and function *J = J(0)*
and in general it is hard to describe all pairs *(E , J)* that produce involutive orbits
however one interesting class of involutive orbits can be outlined by the following theorem:

[E , [E , W]] = 0

and any function
W(dL_{E}J) = c[E , W](dJ) c ∈ ℝ∖(0∪ℕ)

one-parameter family of functions
[E , W](dL_{E}J) + W(d(L_{E})²J) = c[E,[E , W]](dJ) + c[E , W](dL_{E}J)

where
W(d(L_{E})²J) = (c − 1)[E , W](dL_{E}J)

that after
W(d(L_{E})^{m + 1}J) = (c − m)[E , W](d(L_{E})^{m}J)

Now using this property let us prove that functions
{J^{(k)}, J^{(m)}} = W(dJ^{(k)} ∧ dJ^{(m)})

Suppose that
W(dJ^{(k)} ∧ dJ^{(m)}) = W(d(L_{E})^{k}J ∧ dJ^{(m)})
= L_{W(d(LE)kJ)}J^{(m)}

= (c − k + 1)L_{[E , W](d(LE)k − 1J)}J^{(m)}
= (c − k + 1)[E , W](dJ^{(k − 1)} ∧ dJ^{(m)})

= − (c − k + 1)L_{[E , W](d(LE)mJ)}J^{(k − 1)}
= − L_{W(d(LE)m + 1J)}J^{(k − 1)}

= W(dJ^{(k − 1)} ∧ dJ^{(m + 1)})

Thus we have
= (c − k + 1)L

= − (c − k + 1)L

= W(dJ

(c − m){J^{(k)}, J^{(m)}} = (c − k + 1){J^{(k − 1)}, J^{(m + 1)}}

Using this property
{J^{(k)}, J^{(m)}} = {J^{(m)}, J^{(k)}}

and since Poisson bracket is skew-symmetric we finally get
{J^{(k)}, J^{(m)}} = 0

So we showed that functions
S = (c − m)E + t(c − m + 1)W(dJ^{(m + 1)})

is non-Noether symmetry [1] of Hamiltonian dynamical system
f = {J^{(m)}, f}

in other words non-Poisson vector field
X = + W(dJ^{(m)})

This fact can be checked directly
[S , X] = (c − m)[E , X] + t(c − m + 1)[W(dJ^{(m + 1)}), W(dJ^{(m)})]

− (c − m + 1)W(dJ^{(m + 1)}) = (c − m)[E , W](dJ^{(m)}) + (c − m)W(dL_{E}J^{(m)})

+ t(c − m + 1)W(d{J^{(m + 1)},J^{(m)}}) − (c − m + 1)W(dJ^{(m + 1)})

= W(dJ^{(m + 1)}) + (c − m)W(dJ^{(m + 1)}) − (c − m + 1)W(dJ^{(m + 1)}) = 0

In the same time property (9) means that functions
− (c − m + 1)W(dJ

+ t(c − m + 1)W(d{J

= W(dJ

In many infinite dimensional integrable Hamiltonian systems Poisson bivector has nontrivial kernel,
and set of conservation laws belongs to orbit of non-Noether symmetry group that goes through
centre of Poisson algebra. This fact is reflected in the following theorem:

[E, [E , W]] = 0

then every orbit derived from centre
W(dL_{E}J) = − [E , W](dJ)

that according to Theorem 1 ensures involutivity of
u_{t} = cv_{xx} + u_{x}v + uv_{x}

v_{t} = − cu_{xx} + uu_{x} + 3vv_{x}

where v

f = {h, f} = W(dh ∧ df)

with the following Hamiltonian
h = ½ (u²v + v³ + 2cuv_{x})dx

and Poisson bracket defined by Poisson bivector field
W = ½ (A ∧ A_{x} + B ∧ B_{x})dx

where
[E,[E,W]] = 0

this vector field has the following form
E = (uvA_{x} − cvA_{xx} +
(uu_{x} + vv_{x})B + (u² + 2v²)B_{x} + cuB_{xx})xdx

= − [(uv + 2cv_{x} + x((uv)_{x} + cv_{xx}))A

+ (u² + 2v² − 2cu_{x} + x(uu_{x} + 3vv_{x} − cu_{xx}))B]dx

Applying one-parameter group of transformations generated by this vector field to centre of Poisson algebra
which in our case is formed by functional
= − [(uv + 2cv

+ (u² + 2v² − 2cu

J = (ku + mv)dx

where
J^{(0)} = (ku + mv)dx

J^{(1)} = L_{E}J^{(0)}
= (u² + v²)dx

J^{(2)} = (L_{E})²J^{(0)}
= m (u²v + v³ + 2cuv_{x})dx

J^{(3)} = (L_{E})³J^{(0)} =
(u^{4} + 5v^{4} + 6u²v²

− 12cv²u_{x} + 4c²u_{x}² + 4c²v_{x}²)dx

J^{(m)} = (L_{E})^{m}J^{(0)} = L_{E}J^{(m − 1)}

J

J

J

− 12cv²u

J

u_{t} = cu_{xx} + 2uu_{x}

v_{t} = − cv_{xx} + 2uv_{x} + 2u_{x}v

where v

f = {h, f} = W(dh ∧ df)

with the Hamiltonian equal to
h = (u²v + cu_{x}v)dx

and Poisson bracket defined by
W = A ∧ B_{x}dx

One can show that the following vector field
E = (u²A_{x} − cuA_{xx}
+ (uv)_{x}B + 3uvB_{x} + cvB_{xx})xdx

= − [(u² + 2cu_{x} + x(2uu_{x} + cu_{xx}))A

+ (3uv − 2cv_{x} + x(2(uv)_{x} − cv_{xx}))B]dx

has property
= − [(u² + 2cu

+ (3uv − 2cv

[E,[E,W]] = 0

and thus group of transformations generated by this vector field transforms centre of Poisson algebra
formed by functional
J = (ku + mv)dx

into involutive orbit that reproduces well known infinite set of conservation laws
of modified Broer-Kaup hierarchy
J^{(0)} = (ku + mv)dx

J^{(1)} = L_{E}J^{(0)} = m uvdx

J^{(2)} = (L_{E})²J^{(0)}
= 2m (u²v + cu_{x}v)dx

J^{(3)} = (L_{E})³J^{(0)}
= 3m (2u³v − 3cu²v_{x} − 2c²u_{x}v_{x})dx

J^{(m)} = (L_{E})^{m}J^{(0)} = L_{E}J^{(m − 1)}

J

J

J

J

Two samples discussed above are representatives of one interesting family of infinite dimensional
Hamiltonian systems formed by *D* partial differential equations of the following type
*U* is vector with components *u*_{m}
that are smooth functions on *ℝ²* subjected to zero boundary conditions
*G* is constant symmetric nondegenerate matrix, *F* is constant skew-symmetric matrix,
*C* is constants vector that satisfies condition
*〈 · , · 〉* denotes scalar product
*A* is vector with components *A*_{m} that are vector fields defined
for every smooth functional *R(u)* via variational derivatives *A*_{m}(R) = δR/δu_{m}.
Moreover this model is actually bi-Hamiltonian as there exist another invariant Poisson bivector
*W* or in other words
*W*) is formed by
functionals of the following type
*K* is arbitrary constant vector and applying group of transformations generated by *E*
to this functional *J* yields the infinite sequence of functionals
*E* satisfies condition
*J*^{(m)} is involutive.
So *J*^{(m)} are conservation laws of bi-Hamiltonian dynamical system (37)
and vector field *E* is related to non-Noether symmetries of evolutionary equations
(see Remark 1).

U_{t} = − 2FGU_{xx} + 〈U , GU_{x}〉C
+ 〈C , GU_{x}〉U + 〈C , GU〉U_{x}

detG ≠ 0, G^{T} = G, F^{T} = − F

F_{mn}C_{k} + F_{km}C_{n} + F_{nk}C_{m} = 0

where detG ≠ 0, G

F

u_{m} = u_{m}(x, t); u_{m}(±∞, t) = 0; m = 1 ... D

F_{mn}C_{k} + F_{km}C_{n} + F_{nk}C_{m} = 0

and
〈X , Y〉 = X_{m}Y_{m}.

System of equations (37) is Hamiltonian with respect to Poisson bivector equal to
W = 〈A , G^{−1}A_{x}〉dx

where
Ŵ = {〈C , A〉〈U , A_{x}〉 + 〈A_{x} , FA_{x}〉}dx

that is compatible with
[W , W] = [W , Ŵ] = [Ŵ , Ŵ] = 0

Corresponding Hamiltonians that produce Hamiltonian realization
U = Ŵ(dĤ ∧ dU) = W(dH ∧ dU)

of the evolution equations (37) are
Ĥ = ½〈U , GU〉dx

and
H = ½{〈C , GU〉〈U , GU〉 + 2〈FGU_{x} , GU〉}dx

The most remarkable property of system (37) is that it possesses
set of conservation laws that belong to single orbit obtained from
centre of Poisson algebra via one-parameter
group of transformations generated by the following vector field
E = {〈C , GU〉〈U , A_{x}〉 + 〈U , GU〉〈C , A_{x}〉

+ 〈U , GU_{x}〉〈C , A〉 + 2〈FGU , A_{xx}〉}xdx

= {〈C , GU〉〈U , A〉 + 〈U , GU〉〈C , A〉 + 4〈FGU_{x} , A〉

+ x (〈C , GU_{x}〉〈U , A〉 + 〈C , GU〉〈U_{x} , A〉

+ 〈U , GU_{x}〉〈C , A〉 + 2〈FGU_{xx} , A〉)}dx

Note that centre of Poisson algebra (with respect to bracket defined by + 〈U , GU

= {〈C , GU〉〈U , A〉 + 〈U , GU〉〈C , A〉 + 4〈FGU

+ x (〈C , GU

+ 〈U , GU

J = 〈K , U〉dx

where
J^{(0)} = 〈K , U〉dx

J^{(1)} = L_{E}J^{(0)} = ½〈C , K〉〈U , GU〉dx

J^{(2)} = (L_{E})²J^{(0)}
= 〈C , K〉{〈C , GU〉〈U , GU〉 + 2〈FGU_{x} , GU〉}dx

J^{(3)} = (L_{E})³J^{(0)} = ¼〈C , K〉{3〈C , GC〉〈U , GU〉²

+ 12〈C , GU〉²〈U , GU〉 + 32〈C , GU〉〈GU , FGU_{x}〉

+ 24〈U , GC〉〈GU , FGU_{x}〉 + 48〈FGU_{x} , GFGU_{x}〉}dx

J^{(m)} = (L_{E})^{m}J^{(0)} = L_{E}J^{(m − 1)}

One can check that the vector field J

J

J

+ 12〈C , GU〉²〈U , GU〉 + 32〈C , GU〉〈GU , FGU

+ 24〈U , GC〉〈GU , FGU

J

[E , [E , W]] = 0

and according to Theorem 2 the sequence
Note that in special case when *C, F, G, K* have the following form
*C, F, G, K*

D = 2, F_{12} = − F_{21} = ½c, C = K = (0 , 1), G = 1

model (37) reduces to modified Boussinesq system discussed above.
Another choice of constants
D = 2, F_{12} = − F_{21} = ½c, C = K = (0 , 1)

G_{12} = G_{21} = 1, G_{11} = G_{22} = 0

gives rise to Broer-Kaup system described in previous sample.
G

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