We consider set of functions on Poisson manifold related by continues one-parameter group of transformations. Class of vector fields that produce involutive families of functions is investigated and relationship between these vector fields and non-Noether symmetries of Hamiltonian dynamical systems is outlined. Theory is illustrated with sample models: modified Boussinesq system and Broer-Kaup system.

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 ^{∞}(M)^{∞}(M)^{zLE}
^{∞}(M)^{zLE}(J) = J + zL^{2}(L^{2}J + ⋯

Involutivity of orbit

^{zLE}(J)

^{2}J) = c[E,[E , W]](dJ) + c[E , W](dL^{2}J) = (c − 1)[E , W](dL^{m + 1}J) = (c − m)[E , W](d(L^{m}J)
^{(m)} = (L^{m}J^{(k)}, J^{(m)}} = W(dJ^{(k)} ∧ dJ^{(m)})
^{(k)} ∧ dJ^{(m)}) = W(d(L^{k}J ∧ dJ^{(m)})
= L^{k}J)^{(m)}^{k − 1}J)^{(m)}
= (c − k + 1)[E , W](dJ^{(k − 1)} ∧ dJ^{(m)})^{m}J)^{(k − 1)}
= − ^{m + 1}J)^{(k − 1)}^{(k − 1)} ∧ dJ^{(m + 1)})
^{(k)}, J^{(m)}} = (c − k + 1){J^{(k − 1)}, J^{(m + 1)}}
^{(k)}, J^{(m)}} = {J^{(m)}, J^{(k)}}
^{(k)}, J^{(m)}} = 0
^{(m)} = (L^{m}J^{(m)}

Property ^{(m + 1)})
^{(m)}, f}
^{(m)})
^{(m + 1)}), W(dJ^{(m)})] ^{(m + 1)}) = (c − m)[E , W](dJ^{(m)}) + (c − m)W(dL^{(m)}) ^{(m + 1)},J^{(m)}}) − (c − m + 1)W(dJ^{(m + 1)}) ^{(m + 1)}) + (c − m)W(dJ^{(m + 1)}) − (c − m + 1)W(dJ^{(m + 1)}) = 0
^{(m)} = (L^{m}J

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:

^{∞}(M)

^{∞}(M)

^{2}^{2}v + v^{3} + 2cuv^{2} + 2v^{2})B^{2} + 2v^{2} − 2cu^{(0)} = ^{(1)} = L^{(0)}
= ^{2} + v^{2})dx^{(2)} = (L^{2}J^{(0)}
= m^{2}v + v^{3} + 2cuv^{(3)} = (L^{3}J^{(0)} =
^{4} + 5v^{4} + 6u^{2}v^{2}^{2}u^{2}u^{2} + 4c^{2}v^{2})dx^{(m)} = (L^{m}J^{(0)} = L^{(m − 1)}

^{2}^{2}v + cu^{2}A^{2} + 2cu^{(0)} =^{(1)} = L^{(0)} = m^{(2)} = (L^{2}J^{(0)}
= 2m^{2}v + cu^{(3)} = (L^{3}J^{(0)}
= 3m^{3}v − 3cu^{2}v^{2}u^{(m)} = (L^{m}J^{(0)} = L^{(m − 1)}

Two samples discussed above are representatives of one interesting family of infinite dimensional
Hamiltonian systems formed by ^{T} = G, F^{T} = − F^{2}^{−1}A^{(0)} = ^{(1)} = L^{(0)} = ½〈C , K〉^{(2)} = (L^{2}J^{(0)}
= 〈C , K〉^{(3)} = (L^{3}J^{(0)} = ¼〈C , K〉^{2}^{2}〈U , GU〉 + 32〈C , GU〉〈GU , FGU^{(m)} = (L^{m}J^{(0)} = L^{(m − 1)}
^{(m)}^{(m)}

Note that in special case when