_{X}ω = 0 ⇒ X = 0 2-form) and time evolution is governed by Hamilton's equation

_{Xh}ω + dh = 0

_{h}is Hamiltonian vector field that defines time evolution

_{h}(f)

_{Xh}ω denotes contraction of $X$

_{h}and $\omega $. Vector field is said to be (locally) Hamiltonian if it preserves $\omega $. According to the Liouville's theorem $X$

_{h}defined by (1) automatically preserves $\omega $ due to relation

_{Xh}ω = di

_{Xh}ω + i

_{Xh}dω = − ddh = 0

^{aLE}

_{*}(A) = g

_{*}(

_{*}(pdq − hdt) ≠ 0

_{E}(pdq − hdt)) = L

_{E}ω − dE(h) ∧ dt ≠ 0 (first term in r.h.s. does not vanish since $E$ is non-Hamiltonian and as far as $E$ is time independent $L$

_{E}ω and $dE(h)\; \wedge \; dt$ are linearly independent 2-forms). As a result every non-Hamiltonian vector field $E$ commuting with $X$

_{h}leads to the non-Noether symmetry (since $E$ preserves vector field tangent to solutions $L$

_{E}(X

_{h}) = [E , X

_{h}] = 0 it maps the space of solutions onto itself). Any such symmetry yields the following integrals of motion [1],[2],[4],[5]

^{(k)}= Tr(R

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

_{E}ω and $n$ is half-dimension of phase space.

_{E}) carries bi-Hamiltonian structure (§4.12 in [6],[7]-[9]). Indeed $\omega $

_{E}is closed ($d\omega $

_{E}= dL

_{E}ω = L

_{E}dω = 0) and invariant ($L$

_{Xh}ω

_{E}= L

_{Xh}L

_{E}ω = L

_{E}L

_{Xh}ω = 0) 2-form (but generic $\omega $

_{E}is degenerate). So every non-Noether symmetry quite naturally endows dynamical system with bi-Hamiltonian structure.

_{Xh}ω

^{∗}= 0. Let us call it global bi-Hamiltonian structure whenever $\omega \ast $ is exact (there exists 1-form $\theta \ast $ such that $\omega \ast =\; d\theta \ast $) and $X$

_{h}is (globally) Hamiltonian vector field with respect to $\omega \ast $ ($i$

_{Xh}ω

^{∗}+ dh

^{∗}= 0). As far as $\omega $ is nondegenerate there exists vector field $E\ast $ such that $i$

_{E∗}ω = θ

^{∗}. By construction

_{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, i. e., $[X$

_{h}, E] = X

_{h'}. So $E\ast $ is not generator of symmetry since it does not commute with $X$

_{h}but one can construct (locally) Hamiltonian counterpart of $E\ast $ (note that $E\ast $ itself is non-Hamiltonian) — $X$

_{g}with

_{g}is a locally Hamiltonian vector field, satisfying, by construction, the same commutation relations as $E\ast $ (namely $[X$

_{h}, X

_{g}] = X

_{h'}). Finally one recovers 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 global bi-Hamiltonian structure is naturally associated with (non-Noether) symmetry of space of solutions.

_{m}

^{2}ω =

_{m}∧ dq

_{m}

^{∗}=

_{m}dp

_{m}∧ dq

_{m}

_{h}preserves $\omega \ast $. Conserved quantities $p$

_{m}are associated with this simple bi-Hamiltonian structure. This system can be obtained from the following (non-Noether) symmetry (infinitesimal form)

_{m}→ (1 + ap

_{m})q

_{m}

_{m}→ (1 + ap

_{m})p

_{m}

_{t}+ u

_{xxx}+ uu

_{x}= 0

_{Xh}ω+ dh = 0

_{h}=

_{t}

_{x}= u) and the function

^{3}

_{x}

^{2})

_{E}generated by the non-Hamiltonian vector field

_{xx}+

^{2}

_{F}

^{2}v

^{2}⁾

^{3}⁾

_{x}=

^{3}

_{x}

^{2}. The physical origin of this symmetry is unclear, however the symmetry seems to be very important since it leads to the celebrated infinite sequence of conservation laws in involution:

^{(1)}=

^{(2)}=

^{2}dx

^{(3)}=

^{3}

_{x}

^{2}) dx

^{(4)}=

^{4}−

_{x}

^{2}+ u

_{xx}

^{2}) dx

_{Xh∗}ω

^{∗}+ dh

^{∗}= 0

_{E}ω and $h\ast =\; L$

_{E}h) is a result of invariance of KdV under aforementioned transformations $g(a)$.