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 {f , g} = W(df ∧ dg) where 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 gz = e^zLE where LE 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) = gz(J) = e^zLE(J) = J + zLEJ + ½z²(LE)²J + ⋯ the orbit J(z) is called involutive if {J(x) , J(y)} = 0          ∀x, y ∈ ℝ Involutive orbits are often related to integrable models where J(z) plays the role of involutive family of conservation laws.

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:

Theorem 1. For any non-Poisson [E , W] ≠ 0 vector field E satisfying property [E , [E , W]] = 0 and any function J such that W(dLEJ) = c[E , W](dJ)          c ∈ ℝ∖(0∪ℕ) one-parameter family of functions J(z) = e^zLE(J) is involutive.

Proof. By taking Lie derivative of property (6) along the vector field E we get [E , W](dLEJ) + W(d(LE)²J) = c[E,[E , W]](dJ) + c[E , W](dLEJ) where c is real constant which is neither zero nor positive integer. Taking into account (5) one can rewrite result as follows W(d(LE)²J) = (c − 1)[E , W](dLEJ) that after m iterations produces W(d(LE)^{m + 1}J) = (c − m)[E , W](d(LE)^mJ) Now using this property let us prove that functions J^(m) = (LE)^mJ are in involution. Indeed {J^(k), J^(m)} = W(dJ^(k) ∧ dJ^(m)) Suppose that k > m and let us rewrite Poisson bracket as follows W(dJ^(k) ∧ dJ^(m)) = W(d(LE)^kJ ∧ dJ^(m)) = LW(d(LE)^kJ)J^(m) = (c − k + 1)L[E , W](d(LE)^{k − 1}J)J^(m) = (c − k + 1)[E , W](dJ^{(k − 1)} ∧ dJ^(m)) = − (c − k + 1)L[E , W](d(LE)^mJ)J^{(k − 1)} = − c − k + 1c − mLW(d(LE)^{m + 1}J)J^{(k − 1)} = c − k + 1c − mW(dJ^{(k − 1)} ∧ dJ^{(m + 1)}) Thus we have (c − m){J^(k), J^(m)} = (c − k + 1){J^{(k − 1)}, J^{(m + 1)}} Using this property 2(m − k) times produces {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 J^(m) = (LE)^mJ are in involution. In the same time orbit J(z) is linear combination of functions J^(m) and thus it is involutive as well. ∎

Property (9) implies that vector field S = (c − m)E + t(c − m + 1)W(dJ^{(m + 1)}) is non-Noether symmetry [1] of Hamiltonian dynamical system ddtf = {J^(m), f} in other words non-Poisson vector field S commutes with time evolution defined by Hamiltonian vector field X = ∂∂t + 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(dLEJ^(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 J^(m) = (LE)^mJ form Lenard scheme with respect to bi-Hamiltonian structure formed by Poisson bivector fields W and [E , W] (see [1],[4]).

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:

Theorem 2. If non-Poisson vector field E satisfies property [E, [E , W]] = 0 then every orbit derived from centre I of Poisson algebra C^∞(M) is involutive.

Proof. If function J belongs to centre J ∈ I of Poisson algebra C^∞(M) then by definition W(dJ) = 0. By taking Lie derivative of this condition along vector field E one gets W(dLEJ) = − [E , W](dJ) that according to Theorem 1 ensures involutivity of J(z) orbit. ∎

Example. The theorems proved above may have interesting applications in theory of infinite dimensional Hamiltonian models where they provide simple way to construct involutive family of conservation laws. One non-trivial example of such a model is modified Boussinesq system [2],[5],[6] described by the following set of partial differential equations ut = cvxx + uxv + uvx vt = − cuxx + uux + 3vvx where u = u(x, t), v = v(x, t) are smooth functions on ℝ² subjected to zero boundary conditions u(±∞, t) = v(±∞, t) = 0 This system can be rewritten in Hamiltonian form ddtf = {h, f} = W(dh ∧ df) with the following Hamiltonian h = ½∫− ∞+ ∞ (u²v + v³ + 2cuvx)dx and Poisson bracket defined by Poisson bivector field W = ½∫− ∞+ ∞ (A ∧ Ax + B ∧ Bx)dx where A, B are vector fields that for every smooth functional R = R(u) are defined via variational derivatives A(R) = δR/δu and B(R) = δR/δv. For Poisson bivector (24) there exist vector field E such that [E,[E,W]] = 0 this vector field has the following form E = ∫− ∞+ ∞ (uvAx − cvAxx + (uux + vvx)B + (u² + 2v²)Bx + cuBxx)xdx = − ∫− ∞+ ∞ [(uv + 2cvx + x((uv)x + cvxx))A + (u² + 2v² − 2cux + x(uux + 3vvx − cuxx))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 J = ∫− ∞+ ∞ (ku + mv)dx where k, m are arbitrary constants, produces involutive orbit that recovers infinite sequence of conservation laws of modified Boussinesq hierarchy J⁽⁰⁾ = ∫− ∞+ ∞ (ku + mv)dx J⁽¹⁾ = LEJ⁽⁰⁾ = m2∫− ∞+ ∞(u² + v²)dx J⁽²⁾ = (LE)²J⁽⁰⁾ = m∫− ∞+ ∞ (u²v + v³ + 2cuvx)dx J⁽³⁾ = (LE)³J⁽⁰⁾ = 3m4∫− ∞+ ∞ (u⁴ + 5v⁴ + 6u²v² − 12cv²ux + 4c²ux² + 4c²vx²)dx J^(m) = (LE)^mJ⁽⁰⁾ = LEJ^{(m − 1)}

Example. Another interesting model that has infinite sequence of conservation laws lying on single orbit of non-Noether symmetry group is Broer-Kaup system [3],[5],[6], or more precisely special case of Broer-Kaup system formed by the following partial differential equations ut = cuxx + 2uux vt = − cvxx + 2uvx + 2uxv where u = u(x, t), v = v(x, t) are again smooth functions on ℝ² subjected to zero boundary conditions u(±∞, t) = v(±∞, t) = 0 Equations (29) can be rewritten in Hamiltonian form ddtf = {h, f} = W(dh ∧ df) with the Hamiltonian equal to h = ∫− ∞+ ∞ (u²v + cuxv)dx and Poisson bracket defined by W = ∫− ∞+ ∞ A ∧ Bxdx One can show that the following vector field E E = ∫− ∞+ ∞(u²Ax − cuAxx + (uv)xB + 3uvBx + cvBxx)xdx = − ∫− ∞+ ∞ [(u² + 2cux + x(2uux + cuxx))A + (3uv − 2cvx + x(2(uv)x − cvxx))B]dx has property [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⁾ = LEJ⁽⁰⁾ = m∫− ∞+ ∞ uvdx J⁽²⁾ = (LE)²J⁽⁰⁾ = 2m∫− ∞+ ∞ (u²v + cuxv)dx J⁽³⁾ = (LE)³J⁽⁰⁾ = 3m∫− ∞+ ∞ (2u³v − 3cu²vx − 2c²uxvx)dx J^(m) = (LE)^mJ⁽⁰⁾ = LEJ^{(m − 1)}

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 Ut = − 2FGUxx + 〈U , GUx〉C + 〈C , GUx〉U + 〈C , GU〉Ux detG ≠ 0,          G^T = G,       F^T = − F FmnCk + FkmCn + FnkCm = 0 where U is vector with components um that are smooth functions on ℝ² subjected to zero boundary conditions um = um(x, t);          um(±∞, t) = 0;          m = 1 ... D G is constant symmetric nondegenerate matrix, F is constant skew-symmetric matrix, C is constants vector that satisfies condition FmnCk + FkmCn + FnkCm = 0 and 〈 · , · 〉 denotes scalar product 〈X , Y〉 = ∑m=1DXmYm. System of equations (37) is Hamiltonian with respect to Poisson bivector equal to W = ∫− ∞+ ∞〈A , G⁻¹Ax〉dx where A is vector with components Am that are vector fields defined for every smooth functional R(u) via variational derivatives Am(R) = δR/δum. Moreover this model is actually bi-Hamiltonian as there exist another invariant Poisson bivector Ŵ = ∫− ∞+ ∞{〈C , A〉〈U , Ax〉 + 〈Ax , FAx〉}dx that is compatible with W or in other words [W , W] = [W , Ŵ] = [Ŵ , Ŵ] = 0 Corresponding Hamiltonians that produce Hamiltonian realization ddtU = 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〈FGUx , 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 , Ax〉 + 〈U , GU〉〈C , Ax〉 + 〈U , GUx〉〈C , A〉 + 2〈FGU , Axx〉}xdx = ∫− ∞+ ∞{〈C , GU〉〈U , A〉 + 〈U , GU〉〈C , A〉 + 4〈FGUx , A〉 + x (〈C , GUx〉〈U , A〉 + 〈C , GU〉〈Ux , A〉 + 〈U , GUx〉〈C , A〉 + 2〈FGUxx , A〉)}dx Note that centre of Poisson algebra (with respect to bracket defined by W) is formed by functionals of the following type J = ∫− ∞+ ∞〈K , U〉dx where K is arbitrary constant vector and applying group of transformations generated by E to this functional J yields the infinite sequence of functionals J⁽⁰⁾ = ∫− ∞+ ∞〈K , U〉dx J⁽¹⁾ = LEJ⁽⁰⁾ = ½〈C , K〉∫− ∞+ ∞〈U , GU〉dx J⁽²⁾ = (LE)²J⁽⁰⁾ = 〈C , K〉∫− ∞+ ∞{〈C , GU〉〈U , GU〉 + 2〈FGUx , GU〉}dx J⁽³⁾ = (LE)³J⁽⁰⁾ = ¼〈C , K〉∫− ∞+ ∞{3〈C , GC〉〈U , GU〉² + 12〈C , GU〉²〈U , GU〉 + 32〈C , GU〉〈GU , FGUx〉 + 24〈U , GC〉〈GU , FGUx〉 + 48〈FGUx , GFGUx〉}dx J^(m) = (LE)^mJ⁽⁰⁾ = LEJ^{(m − 1)} One can check that the vector field E satisfies condition [E , [E , W]] = 0 and according to Theorem 2 the sequence 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).

Note that in special case when C, F, G, K have the following form D = 2,      F12 = − F21 = ½c,      C = K = (0 , 1),      G = 1 model (37) reduces to modified Boussinesq system discussed above. Another choice of constants C, F, G, K D = 2,     F12 = − F21 = ½c,      C = K = (0 , 1) G12 = G21 = 1,      G11 = G22 = 0 gives rise to Broer-Kaup system described in previous sample.

Summary. Groups of transformations of Poisson manifold that possess involutive orbits play important role in some integrable models where conservation laws form orbit of non-Noether symmetry group. Therefore classification of vector fields that generate such a groups would create good background for description of remarkable class of integrable system that have interesting geometric origin. The present paper is an attempt to outline one particular class of vector fields that are related to non-Noether symmetries of Hamiltonian dynamical systems and produce involutive families of conservation laws.