_{Xh}ω + dh = 0,

_{X}ω = 0 ⇒ X = 0) 2-form, $h$ is the Hamiltonian and $i$

_{X}ω denotes contraction of $X$ with $\omega $. Since $\omega $ is non-degenerate, this gives rise to an isomorphism between the vector fields and 1-forms given by $i$

_{X}ω + α= 0. The vector field is said to be Hamiltonian if it corresponds to exact form

_{Xf}ω + df = 0.

_{f}g = − X

_{g}f = i

_{Xf}i

_{Xg}ω.

_{X}i

_{Y}ω = i

_{W}i

_{X}ω ∧ i

_{Y}ω,

_{W}df ∧ dg.

_{X}i

_{Y}L

_{Z}ω = i

_{[Z,W]}i

_{X}ω ∧ i

_{Y}ω,

^{s}] ∧ U

^{s}+

^{s}∧ [X,U

^{s}]

_{X}i

_{W}ω = i

_{[X,W]}ω + i

_{W}L

_{X}ω.

_{X}i

_{W}ω = L

_{X}

_{Vs ∧ Us}ω = L

_{X}

_{Us}i

_{Vs}ω

_{[X,Us]}i

_{Vs}ω +

_{Us}i

_{[X,Vs]}ω +

_{Us}i

_{Vs}L

_{X}ω = i

_{[X,W]}ω + i

_{W}L

_{X}ω

_{Z}denotes the Lie derivative along the vector field $Z$. According to Liouville's theorem Hamiltonian vector field preserves $\omega $

_{Xf}ω = 0;

_{f},W] = 0.

_{s}where $\omega \; =\sum rs\omega rsdz$

_{r}∧ z

_{s}bivector field $W$ has the following form $W\; =\sum rsWrs\partial \partial z$

_{r}∧

_{s}

_{E}generated by the vector field $E$ maps the space of solutions of equation onto itself if

_{Xh}g

_{*}(ω) + g

_{*}(dh) = 0

_{h}satisfying

_{Xh}ω + dh = 0

_{h}] = 0 Indeed,

_{Xh}L

_{E}ω + dL

_{E}h = L

_{E}(i

_{Xh}ω + dh) = 0

_{h}] = 0. When $E$ is not Hamiltonian, the group of transformations $g(z)\; =\; ezL$

_{E}is non-Noether symmetry (in a sense that it maps solutions onto solutions but does not preserve action).

^{(k)}= i

_{Wk}ω

_{E}

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

_{E}

^{k}are outer powers of $W$ and $L$

_{E}ω.

_{Xh}I

^{(k)}= 0 is fulfilled. Let us consider $L$

_{Xh}I

^{(1)}

_{Xh}I

^{(1)}= L

_{Xh}(i

_{W}ω

_{E}) = i

_{[Xh , W]}ω

_{E}+ i

_{W}L

_{Xh}ω

_{E},

_{h}, W] = 0 and

_{W}L

_{Xh}L

_{E}ω = i

_{W}L

_{E}L

_{Xh}ω = 0

_{h}] = 0 and $L$

_{Xh}ω = 0 vanish. In the same manner one can verify that $L$

_{Xh}I

^{(k)}= 0

_{h}] = X

_{f}where $X$

_{f}is an arbitrary Hamiltonian vector field, then $I(k)$ is still conserved. Such a symmetries map the solutions of the equation $i$

_{Xh}ω + dh = 0 on solutions of

_{Xh}g

_{*}(ω) + d(g

_{*}h + f) = 0

_{Wk}g

_{*}(ω)

^{k}where $g$

_{*}(ω) is transformed $\omega $.

_{h}, E]. Namely it is easy to show by taking the Lie derivative of (15) along vector field $E$ that

^{(k)}, f} = i

_{Wk}ω

^{k}

_{[Xf , E]}

_{s}such that $i$

_{us}ω = 0 s = 1,2 ... n − rank(ω), every Hamiltonian vector field is defined up to linear combination of vectors $u$

_{s}. By identifying $X$

_{f}with $X$

_{f}+

_{s}u

_{s}, we can introduce equivalence class $X$

_{f}

^{∗}(then all $u$

_{s}belong to $0\ast $ ). The bivector field $W$ is also far from being unique, but if $W$

_{1}and $W$

_{2}both satisfy

_{X}i

_{Y}ω = i

_{W1,2}i

_{X}ω ∧ i

_{Y}ω,

_{(W1 − W2)}i

_{X}ω ∧ i

_{Y}ω = 0 ∀X,Y

_{1}− W

_{2}=

_{s}∧ u

_{s}

_{s}are some vector fields and $i$

_{us}ω = 0 (in other words when $W$

_{1}− W

_{2}belongs to the class $0\ast $)

_{h}

^{∗}] = 0

^{∗}commutation relation (generates non-Noether symmetry), then the functions

^{(k)}= i

_{Wk}ω

_{E}

^{k}k = 1...rank(ω)

_{ E}= L

_{E}ω) are constant along trajectories.

_{Xh∗}I

^{(1)}= L

_{Xh∗}(i

_{W}ω

_{E}) = i

_{[Xh∗ , W]}ω

_{E}+ i

_{W}L

_{Xh∗}ω

_{E}= 0

_{h}

^{∗}] = 0

^{∗}and $L$

_{Xh∗}ω = 0. The first one is zero as far as $[X$

_{h}

^{∗}, W

^{∗}] = 0

^{∗}and $[E\; ,\; 0\ast ]\; =\; 0\ast $ are satisfied. So $I(1)$ is conserved. Similarly one can show that $L$

_{Xh}I

^{(k)}= 0 is fulfilled.

_{h}

^{∗}] = X

_{f}

^{∗}

_{0}dx

_{0}+

_{s}dx

_{s}

_{0}= (p

^{2}+ m

^{2})

^{1/2}with vanishing canonical Hamiltonian and degenerate 2-form defined by

_{0}ω =

_{s}dp

_{s}∧ dx

_{0}+ p

_{0}dp

_{s}∧ dx

_{s}).

_{u}ω = 0

_{0}

_{0}

_{s}

_{s}

_{0}x

_{0}

_{0}

_{1}x

_{1}

_{1}

_{n}x

_{n}

_{n}

_{h}

^{∗}] = 0

^{∗}because of $X$

_{h}

^{∗}= 0

^{∗}and $[E,u]\; =\; u$. Corresponding integrals of motion are combinations of momenta:

^{(1)}=

_{s}

^{(2)}=

_{r}p

_{s}

^{(n)}=

_{s}