_{z}= e

^{zLE}

_{E}denotes Lie derivative along the vector field $E$. Action of this group on observables (smooth functions on $M$) is given by expansion

_{z}(f) = e

^{zLE}(f) = f + zL

_{E}f + ½(zL

_{E})

^{2}f + ⋯

_{z}will be called symmetry of Hamiltonian system if it preserves manifold of solutions of Hamilton's equation

_{z}(f) also satisfies it. This happens when $g$

_{z}commutes with time evolution operator

_{z}(f) = g

_{z}(

_{z}does not preserve Poisson bracket structure $[E\; ,\; W]\; \ne \; 0$ then the $g$

_{z}is called non-Noether symmetry. Let us briefly recall some basic features of non-Noether symmetries. First of all if $E$ generates non-Noether symmetry then the $n$ functions

_{k}= i

_{Wk}(L

_{E}ω)

^{k}k = 1,2, ... n

_{k}appear to be in involution $\{Y$

_{k}, Y

_{m}} = 0 while the bivector fields $W$ and $[E\; ,\; W]$ (or in terms of 2-forms $\omega $ and $L$

_{E}ω) form bi-Hamiltonian system (see [1]). Due to this features non-Noether symmetries could be effectively used in construction of conservation laws and bi-Hamiltonian structures.

_{s}= p

_{s}

_{s}= ε(s − 1)e

^{qs − 1 − qs}− ε(n − s)e

^{qs − qs + 1}

_{s}∧ dq

_{s}

_{s}

^{2}+

^{qs − qs + 1}

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

_{s}, q

_{s}satisfying Toda equations (7) $g$

_{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}))

^{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}ω leads to the following 2-form

_{E}ω =

_{s}dp

_{s}∧ dq

_{s}+

^{qs − qs + 1}dq

_{s}∧ q

_{s + 1}+

_{r}∧ dp

_{s}

_{E}ω give rise to bi-Hamiltonian structure of Toda model (compare with [2]). The conservation laws (5) associated with the symmetry reproduce well known set of conservation laws of Toda chain.

_{1}= Y

_{1}=

_{s}

_{2}= ½Y

_{1}

^{2}− Y

_{2}= ½

_{s}

^{2}+

^{qs − qs + 1}

_{3}=

_{1}

^{3}− Y

_{1}Y

_{2}+ Y

_{3}=

_{s}

^{3}+

_{s}+ p

_{s + 1}) e

^{qs − qs + 1}

_{4}= ¼Y

_{1}

^{4}− Y

_{1}

^{2}Y

_{2}+ ½Y

_{2}

^{2}+ Y

_{1}Y

_{3}− Y

_{4}

_{s}

^{4}+

_{s}

^{2}+ 2p

_{s}p

_{s + 1}+ p

_{s + 1}

^{2}) e

^{qs − qs + 1}

^{2(qs − qs + 1)}+

^{qs − qs + 2}

_{m}= (− 1)

^{m}Y

_{m}+ m

^{− 1}

^{k}I

_{m − k}Y

_{k}

_{k},Y

_{m}} = 0. Thus the conservation laws as well as the bi-Hamiltonian structure of the non periodic Toda chain appear to be associated with non-Noether symmetry.

_{t}= i(u

_{xx}+ 2u

^{2}ū)

_{t}= i[E(u)

_{xx}+ 2u

^{2}E(ū) + 4uūE(u)]

_{x}+

_{xx}+ uv + xu

^{2}ū) − t(u

_{xxx}+ 6uūu

_{x})

_{x}= uū). In order to construct conservation laws we also need to know Poisson bracket structure and it appears that invariant Poisson bivector field could be defined if $u$ is subjected to either periodic $u(t,\; -\; \infty )\; =\; u(t,\; +\; \infty )$ or zero $u(t,\; -\; \infty )\; =\; u(t,\; +\; \infty )\; =\; 0$ boundary conditions. In terms of variational derivatives the explicit form of the Poisson bivector field is

_{t}= {h , u}

^{2}ū

^{2}− u

_{x}ū

_{x})

_{E}ω =

_{x}∧ δū + uδv ∧ δū + ūδv ∧ δu]dx

_{E}ω exactly reproduces the bi-Hamiltonian structure of NSE proposed by Magri [4] while the conservation laws associated with this symmetry are well known conservation laws of NSE

_{1}= Y

_{1}= 2

_{2}= Y

_{1}

^{2}− 2Y

_{2}= i

_{x}u − u

_{x}ū) dx

_{3}= Y

_{1}

^{3}− 3Y

_{1}Y

_{2}+ 3Y

_{3}= 2

^{2}ū

^{2}− u

_{x}ū

_{x}) dx

_{4}= Y

_{1}

^{4}− 4Y

_{1}

^{2}Y

_{2}+ 2Y

_{2}

^{2}+ 4Y

_{1}Y

_{3}− 4Y

_{4}

_{x}u

_{xx}− u

_{x}ū

_{xx}) + 3i(ūu

^{2}ū

_{x}− uū

^{2}u

_{x})] dx

_{m}= (− 1)

^{m}mY

_{m}+

^{k}I

_{m − k}Y

_{k}

_{k}, Y

_{m}} = 0 is related to the fact that $E$ satisfies Yang-Baxter equation $[[E[E\; ,\; W]]W]\; =\; 0$.

_{t}+ u

_{xxx}+ uu

_{x}= 0 [KdV]

_{t}+ u

_{xxx}− 6u

^{2}u

_{x}= 0 [mKdV]

_{t}+ E(u)

_{xxx}+ u

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

_{x}= 0 [KdV]

_{t}+ E(u)

_{xxx}− 12uu

_{x}E(u) − 6u

^{2}E(u)

_{x}= 0 [mKdV]

_{xx}+

^{2}+

_{x}v +

_{xxx}+ uu

_{x})

_{xxxxx}+ 20u

_{x}u

_{xx}+ 10 uu

_{xxx}+ 5u

^{2}u

_{x}) [KdV]

_{xx}+ 2u

^{3}+ u

_{x}w −

_{xxx}− 6u

^{2}u

_{x})

_{xxxxx}− 10u

^{2}u

_{xxx}− 40uu

_{x}u

_{xx}− 10u

_{x}

^{3}+ 30u

^{4}u

_{x}) [mKdV]

_{x}= u and $w$

_{x}= u

^{2}) To construct conservation laws we need to know Poisson bracket structure and again like in the case of NSE the Poisson bivector field is well defined when $u$ is subjected to either periodic $u(t,\; -\; \infty )\; =\; u(t,\; +\; \infty )$ or zero $u(t,\; -\; \infty )\; =\; u(t,\; +\; \infty )\; =\; 0$ boundary conditions. For both KdV and mKdV the Poisson bivector field is

_{t}= {h , u}

_{x}

^{2}−

^{3}

_{x}

^{2}+ u

^{4}) dx [mKdV]

_{E}ω =

_{x}+

_{E}ω =

_{x}− 2uδu ∧ δw) [mKdV]

_{1}= Y

_{1}=

_{2}= Y

_{1}− 2Y

_{2}=

^{2}dx

_{3}= Y

_{1}

^{3}− 3Y

_{1}Y

_{2}+ 3Y

_{3}=

^{3}

_{x}

^{2}) dx

_{4}= Y

_{1}

^{4}− 4Y

_{1}

^{2}Y

_{2}+ 2Y

_{2}

^{2}+ 4Y

_{1}Y

_{3}− 4Y

_{4}=

^{4}−

_{x}

^{2}+ u

_{xx}

^{2}) dx

_{m}= (− 1)

^{m}mY

_{m}+

^{k}I

_{m − k}Y

_{k}

_{1}= Y

_{1}= − 4

^{2}dx

_{2}= Y

_{1}− 2Y

_{2}= 16

^{4}+ u

_{x}

^{2}) dx

_{3}= Y

_{1}

^{3}− 3Y

_{1}Y

_{2}+ 3Y

_{3}= − 32

^{6}+ 10 u

^{2}u

_{x}

^{2}+ u

_{xx}

^{2}) dx

_{4}= Y

_{1}

^{4}− 4Y

_{1}

^{2}Y

_{2}+ 2Y

_{2}

^{2}+ 4Y

_{1}Y

_{3}− 4Y

_{4}=

^{8}+ 70u

^{4}u

_{x}

^{2}− 7u

_{x}

^{4}+ 14u

^{2}u

_{xx}

^{2}+ u

_{xxx}

^{2}) dx

_{m}= (− 1)

^{m}mY

_{m}+

^{k}I

_{m − k}Y

_{k}