_{z}= e

^{zLE}(here $L$ denotes Lie derivative) that acts on the observables as follows

_{z}(Ô) = e

^{zLE}(Ô) = f + zL

_{E}Ô + ½(zL

_{E})

^{2}Ô + ⋯

_{z}(Ô) = g

_{z}(

_{z}commutes with the vector field $W(h)\; =\; \{h\; ,\; \}$, i. e.

_{z}is called non-Noether symmetry.

^{(k)}=

^{k}∧ W

^{n − k}= Y

^{(k)}W

^{n}.

^{k}∧ W

^{n − k}= (

^{(k)})W

^{n}+ Y

^{(k)}[W(h) , W

^{n}]

^{k − 1}∧ W

^{n − k}

^{k}∧ W

^{n − k − 1}=

^{(k)})W

^{n}+ nY

^{(k)}[W(h) , W] ∧ W

^{n − 1}

^{(k)}W

^{n}= 0

_{1}... c

_{n}of the secular equation

^{n}= 0

_{1}... c

_{n}in the following way

^{(k)}=

_{m[1]}c

_{m[2]}⋯ c

_{m[k]}

_{1}, z

_{2}, z

_{3}, z

_{4}and Poisson bivector field

_{1}∧ D

_{3}+ D

_{2}∧ D

_{4}

_{m}just denotes derivative with respect to $z$

_{m}coordinate) and let's take

_{1}

^{2}+ ½z

_{2}

^{2}+ e

^{z3 − z4}

_{m}D

_{m}

_{1}= ½z

_{1}

^{2}− e

^{z3 − z4}−

_{1}+ z

_{2})e

^{z3 − z4}

_{2}= ½z

_{2}

^{2}+ 2e

^{z3 − z4}+

_{1}+ z

_{2})e

^{z3 − z4}

_{3}= 2z

_{1}+ ½z

_{2}+

_{1}

^{2}+ e

^{z3 − z4})

_{4}= z

_{2}− ½z

_{1}+

_{2}

^{2}+ e

^{z3 − z4})

_{1}D

_{1}∧ D

_{3}+ z

_{2}D

_{2}∧ D

_{4}+ e

^{z3 − z4}D

_{1}∧ D

_{2}+ D

_{3}∧ D

_{4}

_{1}∧ D

_{2}∧ D

_{3}∧ D

_{4}

_{1}+ z

_{2})D

_{1}∧ D

_{2}∧ D

_{3}∧ D

_{4}

_{1}z

_{2}− e

^{z3 − z4}) D

_{1}∧ D

_{2}∧ D

_{3}∧ D

_{4}

^{(1)}=

_{1}+ z

_{2})

^{(2)}=

_{1}z

_{2}− e

^{z3 − z4}

^{(k)}= Tr(L

^{k})

_{m}, where the bivector field $W$ and the generator of the symmetry $E$ have the following form

_{ab}D

_{k}∧ D

_{m}E =

_{m}D

_{m}

_{ab}=

^{−1})

_{ad}(E

_{c}D

_{c}W

_{db}− W

_{cb}D

_{c}E

_{d}+ W

_{dc}D

_{c}E

_{b})

_{ab}=

_{a}(W

_{bc}D

_{c}h)

_{E}(u) = Φ

^{− 1}([E , Φ(u)]) − L

_{E}u

_{E}is a linear operator and it is invariant since time evolution commutes with both $\Phi $ (as far as $[W(h)\; ,\; W]\; =\; 0$) and $E$ (because $E$ generates symmetry). In the terms of the local coordinates $\u0154$

_{E}has the following form

_{E}=

_{ab}dz

_{a}⊗ D

_{b}

_{E}= L

_{W(h)}Ŕ

_{E}= 0

_{E}=

_{ab}dz

_{a}⊗ D

_{b}

_{ab}) dz

_{a}⊗ D

_{b}+

_{ab}(L

_{W(h)}dz

_{a}) ⊗ D

_{b}

_{ab}dz

_{a}⊗ (L

_{W(h)}D

_{b}) =

_{ab}) dz

_{a}⊗ D

_{b}

_{ab}D

_{c}(W

_{ad}D

_{d}h)dz

_{c}⊗ D

_{b}+

_{ab}D

_{b}(W

_{cd}D

_{d}h)dz

_{a}⊗ D

_{c}

_{ab}+

_{ac}L

_{cb}− L

_{ac}P

_{cb}))dz

_{a}⊗ D

_{b}= 0

_{i}in quite simple way:

^{(k)}= Tr(L

^{k}) =

_{m}

^{k}

_{E}(31).

_{11}= L

_{33}= z

_{1}; L

_{22}= L

_{44}= z

_{2}

_{14}= − L

_{23}= e

^{z3 − z4}; L

_{32}= − L

_{41}= 1

^{(1)}= Tr(L) = 2(z

_{1}+ z

_{2})

^{(2)}= Tr(L

^{2}) = 2z

_{1}

^{2}+ 2z

_{2}

^{2}+ 4e

^{z3 − z4}

^{(k)}, Y

^{(m)}} = 0

_{W(f)}W = [W(f) , W] = 0

_{E}[W , W] = [E[W , W]] = [[E , W] W] + [W[E , W]] = 2[Ŵ , W] = 0.

_{i}≠ c

_{j}of the equation (18). By taking the Lie derivative of the equation

_{i}W)

^{n}= 0

_{j}) and $\u0174(c$

_{j}) and using Liouville theorem for $W$ and $\u0174$ bivectors we obtain the following relations

_{i}W)

^{n − 1}(L

_{W(cj)}Ŵ − {c

_{j}, c

_{i}}W) = 0,

_{i}W)

^{n − 1}(c

_{i}L

_{Ŵ(cj)}W + {c

_{j}, c

_{i}}

_{•}W) = 0,

_{i}, c

_{j}}

_{•}= Ŵ(dc

_{i}∧ dc

_{j})

_{i}subtracting (49) and using identity (43) gives rise to

_{i}, c

_{j}}

_{•}− c

_{i}{c

_{i}, c

_{j}})(Ŵ − c

_{i}W)

^{n − 1}W = 0

_{i}, c

_{j}}

_{•}− c

_{i}{c

_{i}, c

_{j}} = 0

_{i}W)

^{n − 1}W vanishes. In the second case we can repeat (48)-(51) procedure for the volume field $(\u0174\; -\; c$

_{i}W)

^{n − 1}W yielding after $n$ iterations $Wn=\; 0$ that according to our assumption (that the dynamical system is regular) is not true. As a result we arrived at (52) and by the simple interchange of indices $i\; \leftrightarrow \; j$ we get

_{i}, c

_{j}}

_{•}− c

_{j}{c

_{i}, c

_{j}} = 0

_{i}are in involution with respect to the both Poisson structures (since $c$

_{i}≠ c

_{j})

_{i}, c

_{j}}

_{•}= {c

_{i}, c

_{j}} = 0

^{(k)}

^{(k)}→ Ω

^{(k + 1)}

^{− 1}([W , Φ(u)])

^{− 1}([[E , W]Φ(u)])

^{2}u = Φ

^{− 1}([W , Φ(Φ

^{− 1}([W , Φ(u)]))])

^{− 1}([W[W , Φ(u)]]) = 0

^{2}u = Φ

^{− 1}([[W , E][[W , E]Φ(u)]]) = 0

^{− 1}([[[W , E]W]Φ(u)]) = 0

^{(k)}= kdI

^{(k + 1)}

_{1}= z

_{1}dz

_{1}− e

^{z3 − z4}dz

_{4}

_{2}= z

_{2}dz

_{2}+ e

^{z3 − z4}dz

_{3}

_{3}= z

_{1}dz

_{3}+ dz

_{2}

_{4}= z

_{2}dz

_{4}− dz

_{1}

^{(1)}= dI

^{(2)}

_{E}(31) and to show how Frölicher-Nijenhuis geometry could arise in Hamiltonian system that possesses certain non-Noether symmetry. From the geometric properties of the tangent valued forms we know that the traces of powers of a linear operator $F$ on tangent bundle are in involution whenever its Frölicher-Nijenhuis torsion $T(F)$ vanishes, i. e. whenever for arbitrary vector fields $X,Y$ the condition

_{E}(X) = Φ(L

_{E}Φ

^{− 1}(X)) − [E , X]

_{E}follows from the invariance of the $\u0154$

_{E}defined by (31) (note that for arbitrary 1-form vector field $u$ and vector field $X$ contraction $i$

_{X}u has the property $i$

_{REX}u = i

_{X}Ŕ

_{E}u, so $R$

_{E}is actually transposed to $\u0154$

_{E}). It remains to show that the condition (39) ensures vanishing of the Frölicher-Nijenhuis torsion $T(R$

_{E}) of $R$

_{E}, i.e. for arbitrary vector fields $X,\; Y$ we must get

_{E})(X , Y) = [R

_{E}(X) , R

_{E}(Y)]

_{E}([R

_{E}(X) , Y] + [X , R

_{E}(Y)] − R

_{E}([X , Y])) = 0

^{− 1}(W), ω

^{•}= Ŕ

_{E}ω ω

^{••}= Ŕ

_{E}ω

^{•}

^{− 1}([W , W]) = 0

^{•}= dΦ

^{− 1}([E , W]) − dL

_{E}ω = Φ

^{− 1}([[E , W]W]) − L

_{E}dω = 0

^{••}= 2Φ

^{− 1}([[E[E , W]]W]) − 2dL

_{E}ω

^{•}= − 2L

_{E}dω

^{•}= 0

^{•}= dω

^{••}= 0

_{E}) and $\omega $.

_{T(RE)(X , Y)}ω = i

_{[REX , REY]}ω − i

_{[REX , Y]}ω

^{•}− i

_{[X , REY]}ω

^{•}+ i

_{[X , Y]}ω

^{••}

_{REX}i

_{Y}ω

^{•}− i

_{REY}L

_{X}ω

^{•}− L

_{REX}i

_{Y}ω

^{•}+ i

_{Y}L

_{REX}ω

^{•}− L

_{X}i

_{REY}ω

^{•}

_{REY}L

_{X}ω

^{•}+ i

_{[X , Y]}ω

^{••}= i

_{Y}L

_{X}ω

^{••}− L

_{X}i

_{Y}ω

^{••}+ i

_{[X , Y]}ω

^{••}= 0

_{X}i

_{Y}ω = i

_{Y}L

_{X}ω + i

_{[X , Y]}ω

_{REX}ω = di

_{REX}ω + i

_{REX}dω = di

_{X}ω

^{•}= L

_{X}ω

^{•}− i

_{X}dω

^{•}= L

_{X}ω

^{•}

_{E})(X , Y) and $\omega $ vanishes. But since $W$ bivector is non-degenerate ($Wn\ne \; 0$), its counter image

^{− 1}(W)

_{E}) itself is zero. So we get

_{E})(X , Y) = [R

_{E}(X) , R

_{E}(Y)]

_{E}([R

_{E}(X) , Y] + [X , R

_{E}(Y)] − R

_{E}([X , Y])) = 0

_{E}associated with non-Noether symmetry (23) reproduces well known Frölicher-Nijenhuis operator

_{E}= z

_{1}dz

_{1}⊗ D

_{1}− dz

_{1}⊗ D

_{4}+ z

_{2}dz

_{2}⊗ D

_{2}+ dz

_{2}⊗ D

_{3}+ z

_{1}dz

_{3}⊗ D

_{3}

^{z3 − z4}dz

_{3}⊗ D

_{2}+ z

_{2}dz

_{4}⊗ D

_{4}− e

^{z3 − z4}dz

_{4}⊗ D

_{1}