Department of Theoretical Physics,
A. Razmadze Institute of Mathematics,
1 Aleksidze Street, Tbilisi 0193, Georgia

We disscuss some geometric aspects of the concept of non-Noether symmetry.
It is shown that in regular Hamiltonian systems such a symmetry canonically leads
to a Lax pair on the algebra of linear operators on cotangent bundle over the phase space.
Correspondence between the non-Noether symmetries and other wide spread geometric
methods of generating conservation laws such as bi-Hamiltonian formalism,
bidifferential calculi and Frölicher-Nijenhuis geometry is considered.
It is proved that the integrals of motion associated with the
continuous non-Noether symmetry are in involution whenever the
generator of the symmetry satisfies a certain Yang-Baxter type equation.

Non-Noether symmetry; Conservation law; bi-Hamiltonian system; Bidifferential calculus; Lax pair; Frölicher-Nijenhuis
operator;

70H33; 70H06; 53Z05

J. Geom. Phys. 48 (2003) 190-202

In the present paper we would like to shed more light on geometric aspects of
the concept of non-Noether symmetry and to emphasize influence of such a symmetries on the phase space geometry.
Partially the motivation for studying these issues comes from the theory of integrable models
that essentially relies on different geometric objects used for constructing conservation
laws. Among them are Frölicher-Nijenhuis
operators, bi-Hamiltonian systems, Lax pairs and bicomplexes. And it seems that the existance of these important
geometric structures could be related to the hidden non-Noether symmetries of the dynamical systems.
We would like to show how in Hamiltonian systems presence of certain non-Noether symmetries leads to the
above mentioned Lax pairs, Frölicher-Nijenhuis operators, bi-Hamiltonian structures,
bicomplexes and a number of conservation laws.

Let us first recall some basic knowledge of the Hamiltonian dynamics. The phase space of
a regular Hamiltonian system is a Poisson manifold – a smooth finite-dimensional
manifold equipped with the Poisson bivector field $W$
subjected to the following condition
$$[W,W]=0$$
where square bracket stands for Schouten bracket or supercommutator
(for simplicity further it will be referred as commutator). In a standard manner Poisson
bivector field defines a Lie bracket on the algebra of observables
(smooth real-valued functions on phase space) called Poisson bracket:
$$\{f,g\}=W(df\wedge dg)$$
Skew symmetry of the bivector field $W$ provides the skew symmetry of
the corresponding Poisson bracket and the condition
(1) ensures that for every triple $(f,g,h)$ of smooth
functions on the phase space the Jacobi identity
$$\left\{f\right\{g,h\left\}\right\}+\left\{h\right\{f,g\left\}\right\}+\left\{g\right\{h,f\left\}\right\}=0.$$
is satisfied. We also assume that the dynamical system under consideration
is regular – the bivector field $W$ has maximal
rank, i. e. its $n$-th outer power, where $n$ is a half-dimension of
the phase space, does not vanish ${W}^{n}\ne 0$.
In this case $W$ gives rise to a well known isomorphism
$\Phi $ between the differential 1-forms and
the vector fields defined by
$$\Phi \left(u\right)=W\left(u\right)$$
for every 1-form $u$ and could be extended to higher degree
differential forms and multivector fields by linearity and multiplicativity
$\Phi (u\wedge v)=\Phi \left(u\right)\wedge \Phi \left(v\right)$.

Time evolution of observables (smooth functions on phase space) is governed by the Hamilton's equation
$$\frac{d}{dt}\xd4=\{h,\xd4\}$$
where $h$ is some fixed smooth function on the phase space called Hamiltonian.
Let us recall that each vector field $E$ on the phase space generates
the one-parameter continuous group of transformations
${g}_{z}={e}^{z{L}_{E}}$ (here $L$ denotes Lie derivative)
that acts on the observables as follows
$${g}_{z}\left(\xd4\right)={e}^{z{L}_{E}}\left(\xd4\right)=f+z{L}_{E}\xd4+\mathrm{\xbd}(z{L}_{E}{)}^{2}\xd4+\cdots $$
Such a group of transformation is called symmetry of Hamilton's equation (5)
if it commutes with time evolution operator
$$\frac{d}{dt}{g}_{z}\left(\xd4\right)={g}_{z}\left(\frac{d}{dt}\xd4\right)$$
in terms of the vector fields this condition means that the generator
$E$ of the group ${g}_{z}$ commutes with the vector field
$W\left(h\right)=\{h,\}$, i. e.
$$[E,W(h\left)\right]=0.$$ However we would like to consider more general
case where $E$ is time dependent vector field on phase space. In this case
(8) should be replaced with
$$\frac{\partial}{\partial t}E=[E,W(h\left)\right].$$
If in addition to (8) the vector field $E$ does not preserve Poisson
bivector field $[E,W]\ne 0$ then ${g}_{z}$ is called non-Noether symmetry.

Now let us focus on non-Noether symmetries. We would like to show that the presence of
such a symmetry could essentially enrich the geometry of the phase space
and under the certain conditions could ensure integrability of the dynamical system.
Before we proceed let us recall that the non-Noether symmetry leads to a number of
integrals of motion
[4]. More precisely the
relationship between non-Noether symmetries and the conservation laws is described by
the following theorem.

Let $(M,h)$ be regular Hamiltonian system on the $2n$-dimensional
Poisson manifold $M$. Then, if the vector field $E$ generates
non-Noether symmetry, the functions
$${Y}^{\left(k\right)}=\frac{V\left[k\right]}{V\left[0\right]}\text{}k=1,2,...n$$
where $V\left[k\right]={\u0174}^{k}\wedge {W}^{n-k}$
are multivector fields of maximal degree constructed by means of Poisson bivector $W$
and its Lie derivative $\u0174=[E,W]$, are integrals of motion.

By the definition
$${\u0174}^{k}\wedge {W}^{n-k}={Y}^{\left(k\right)}{W}^{n}.$$
(definition is correct since the space of $2n$ degree multivector fields on $2n$
degree manifold is one dimensional).
Let us take time derivative of this expression along the vector field $W\left(h\right)$,
$$\frac{d}{dt}{\u0174}^{k}\wedge {W}^{n-k}=\left(\frac{d}{dt}{Y}^{\left(k\right)}\right){W}^{n}+{Y}^{\left(k\right)}\left[W\right(h),{W}^{n}]$$
or
$$k\left(\frac{d}{dt}\u0174\right)\wedge {\u0174}^{k-1}\wedge {W}^{n-k}\phantom{\rule{0ex}{0ex}}+(n-k)\left[W\right(h),W]\wedge {\u0174}^{k}\wedge {W}^{n-k-1}=\phantom{\rule{0ex}{0ex}}\left(\frac{d}{dt}{Y}^{\left(k\right)}\right){W}^{n}+n{Y}^{\left(k\right)}\left[W\right(h),W]\wedge {W}^{n-1}$$
but according to the Liouville theorem the Hamiltonian vector field preserves $W$ i. e.
$$\frac{d}{dt}W=\left[W\right(h),W]=0$$
hence, by taking into account that
$$\frac{d}{dt}E=\frac{\partial}{\partial t}E+\left[W\right(h),E]=0$$ we get
$$\frac{d}{dt}\u0174=\frac{d}{dt}[E,W]=[\frac{d}{dt}E,W]+\left[E\right[W\left(h\right),W\left]\right]=0.$$
and as a result (13) yields
$$\frac{d}{dt}{Y}^{\left(k\right)}{W}^{n}=0$$
but since the dynamical system is regular (${W}^{n}\ne 0$)
we obtain that the functions ${Y}^{\left(k\right)}$ are integrals of motion.

Instead of conserved quantities
${Y}^{\left(1\right)}...{Y}^{\left(n\right)}$, the
solutions ${c}_{1}...{c}_{n}$ of the secular equation
$$(\u0174-cW{)}^{n}=0$$
could be associated with the generator of symmetry.
By expanding expression (18) it is easy to verify that the conservation laws
${Y}^{\left(k\right)}$ can be expressed in terms of the integrals of motion
${c}_{1}...{c}_{n}$ in the following way
$${Y}^{\left(k\right)}=\frac{(n-k)!k!}{n!}\sum _{m\left[i\right]\text{>}m\left[j\right]}{c}_{m\left[1\right]}{c}_{m\left[2\right]}\cdots {c}_{m\left[k\right]}$$

Let $M$ be ${R}^{4}$ with coordinates
${z}_{1},{z}_{2},{z}_{3},{z}_{4}$ and Poisson bivector field
$$W={D}_{1}\wedge {D}_{3}+{D}_{2}\wedge {D}_{4}$$
(${D}_{m}$ just denotes derivative with respect to ${z}_{m}$ coordinate)
and let's take
$$h=\mathrm{\xbd}{z}_{1}^{2}+\mathrm{\xbd}{z}_{2}^{2}+{e}^{{z}_{3}-{z}_{4}}$$
Then the vector field
$$E=\sum _{m=1}^{4}{E}_{m}{D}_{m}$$
with components
$${E}_{1}=\mathrm{\xbd}{z}_{1}^{2}-{e}^{{z}_{3}-{z}_{4}}-\frac{t}{2}({z}_{1}+{z}_{2}){e}^{{z}_{3}-{z}_{4}}\phantom{\rule{0ex}{0ex}}{E}_{2}=\mathrm{\xbd}{z}_{2}^{2}+2{e}^{{z}_{3}-{z}_{4}}+\frac{t}{2}({z}_{1}+{z}_{2}){e}^{{z}_{3}-{z}_{4}}\phantom{\rule{0ex}{0ex}}{E}_{3}=2{z}_{1}+\mathrm{\xbd}{z}_{2}+\frac{t}{2}({z}_{1}^{2}+{e}^{{z}_{3}-{z}_{4}})\phantom{\rule{0ex}{0ex}}{E}_{4}={z}_{2}-\mathrm{\xbd}{z}_{1}+\frac{t}{2}({z}_{2}^{2}+{e}^{{z}_{3}-{z}_{4}})$$
satisfies (9) condition and as a result generates symmetry of the dynamical system.
The symmetry appears to be non-Noether with Schouten bracket $[E,W]$ equal to
$$\u0174=[E,W]={z}_{1}{D}_{1}\wedge {D}_{3}+{z}_{2}{D}_{2}\wedge {D}_{4}+{e}^{{z}_{3}-{z}_{4}}{D}_{1}\wedge {D}_{2}+{D}_{3}\wedge {D}_{4}$$
calculating volume vector fields
${\u0174}^{k}\wedge {W}^{n-k}$ gives rise to
$$W\wedge W=-2{D}_{1}\wedge {D}_{2}\wedge {D}_{3}\wedge {D}_{4}\phantom{\rule{0ex}{0ex}}\u0174\wedge W=-({z}_{1}+{z}_{2}){D}_{1}\wedge {D}_{2}\wedge {D}_{3}\wedge {D}_{4}\phantom{\rule{0ex}{0ex}}\u0174\wedge \u0174=-2({z}_{1}{z}_{2}-{e}^{{z}_{3}-{z}_{4}}){D}_{1}\wedge {D}_{2}\wedge {D}_{3}\wedge {D}_{4}$$
and the conservation laws associated with this symmetry are just
$${Y}^{\left(1\right)}=\frac{\u0174\wedge W}{W\wedge W}=\mathrm{\xbd}({z}_{1}+{z}_{2})\phantom{\rule{0ex}{0ex}}{Y}^{\left(2\right)}=\frac{\u0174\wedge \u0174}{W\wedge W}={z}_{1}{z}_{2}-{e}^{{z}_{3}-{z}_{4}}$$

Presence of the non-Noether symmetry not only leads to a sequence of conservation laws, but also
endows the phase space with a number of interesting geometric structures and it appears that such a
symmetry is related to many important concepts used in theory of dynamical systems.
One of the such concepts is Lax pair.
Let us recall that Lax pair of Hamiltonian system on Poisson manifold $M$ is
a pair $(L,P)$ of smooth functions on $M$ with values in some
Lie algebra $g$ such that the time evolution of $L$ is governed
by the following equation
$$\frac{d}{dt}L=[L,P]$$
where $[,]$ is a Lie bracket on $g$. It is well known that each Lax
pair leads to a number of conservation laws. When $g$ is some matrix Lie algebra
the conservation laws are just traces of powers of $L$
$${I}^{\left(k\right)}=Tr\left({L}^{k}\right)$$
It is remarkable that each generator of the non-Noether
symmetry canonically leads to the Lax pair of a certain type.
In the local coordinates ${z}_{m}$, where the bivector field
$W$ and the generator of the symmetry $E$ have the
following form
$$W=\sum _{km}{W}_{ab}{D}_{k}\wedge {D}_{m}\text{}E=\sum _{m}{E}_{m}{D}_{m}$$
corresponding Lax pair could be calculated explicitly.
Namely we have the following theorem:

Let $(M,h)$ be regular Hamiltonian system on the $2n$-dimensional
Poisson manifold $M$.
Then, if the vector field $E$ on $M$ generates the non-Noether symmetry,
the following $2n\times 2n$ matrix valued functions on $M$
$${L}_{ab}=\sum _{dc}\left({W}^{-1}{)}_{ad}\right({E}_{c}{D}_{c}{W}_{db}-{W}_{cb}{D}_{c}{E}_{d}+{W}_{dc}{D}_{c}{E}_{b})\phantom{\rule{0ex}{0ex}}{P}_{ab}=\sum _{c}{D}_{a}({W}_{bc}{D}_{c}h)$$
form the Lax pair (27) of the dynamical system $(M,h)$.

Let us consider the following operator on a space of 1-forms
$${\text{\u0154}}_{E}\left(u\right)={\Phi}^{-1}\left(\right[E,\Phi \left(u\right)\left]\right)-{L}_{E}u$$
(here $\Phi $ is the isomorphism (4)).
It is obvious that ${\text{\u0154}}_{E}$
is a linear operator and it is invariant
since time evolution commutes with both
$\Phi $
(as far as $\left[W\right(h),W]=0$) and $E$ (because $E$ generates
symmetry). In the terms of the local coordinates ${\text{\u0154}}_{E}$ has the following form
$${\text{\u0154}}_{E}=\sum _{ab}{L}_{ab}d{z}_{a}\otimes {D}_{b}$$
and the invariance condition
$$\frac{d}{dt}{\text{\u0154}}_{E}={L}_{W\left(h\right)}{\text{\u0154}}_{E}=0$$
yields
$$\frac{d}{dt}{\text{\u0154}}_{E}=\frac{d}{dt}\sum _{ab}{L}_{ab}d{z}_{a}\otimes {D}_{b}\phantom{\rule{0ex}{0ex}}=\sum _{ab}\left(\frac{d}{dt}{L}_{ab}\right)d{z}_{a}\otimes {D}_{b}+\sum _{ab}{L}_{ab}\left({L}_{W\left(h\right)}d{z}_{a}\right)\otimes {D}_{b}\phantom{\rule{0ex}{0ex}}+\sum _{ab}{L}_{ab}d{z}_{a}\otimes \left({L}_{W\left(h\right)}{D}_{b}\right)=\sum _{ab}\left(\frac{d}{dt}{L}_{ab}\right)d{z}_{a}\otimes {D}_{b}\phantom{\rule{0ex}{0ex}}+\sum _{abcd}{L}_{ab}{D}_{c}\left({W}_{ad}{D}_{d}h\right)d{z}_{c}\otimes {D}_{b}+\sum _{abcd}{L}_{ab}{D}_{b}\left({W}_{cd}{D}_{d}h\right)d{z}_{a}\otimes {D}_{c}\phantom{\rule{0ex}{0ex}}=\sum _{ab}(\frac{d}{dt}{L}_{ab}+\sum _{c}({P}_{ac}{L}_{cb}-{L}_{ac}{P}_{cb}\left)\right)d{z}_{a}\otimes {D}_{b}=0$$
or in matrix notations
$$\frac{d}{dt}L=[L,P].$$
So, we have proved that the non-Noether symmetry canonically yields a Lax pair
on the algebra of linear operators on cotangent bundle over the phase space.

The conservation laws (28)
associated with the Lax pair (27) can be expressed in terms of the
integrals of motion ${c}_{i}$ in quite simple way:
$${I}^{\left(k\right)}=Tr\left({L}^{k}\right)=\sum _{m}{c}_{m}^{k}$$
This correspondence follows from the equation (18)
and the definition of the operator ${\text{\u0154}}_{E}$ (31).

Let us calculate Lax matrix associated with non-Noether symmetry (23).
Using (30) it is easy to check that Lax matrix has eight nonzero elements
$${L}_{11}={L}_{33}={z}_{1};\text{}{L}_{22}={L}_{44}={z}_{2}\phantom{\rule{0ex}{0ex}}{L}_{14}=-{L}_{23}={e}^{{z}_{3}-{z}_{4}};\text{}{L}_{32}=-{L}_{41}=1$$
The conservation laws associated with this Lax matrix are
$${I}^{\left(1\right)}=Tr\left(L\right)=2({z}_{1}+{z}_{2})\phantom{\rule{0ex}{0ex}}{I}^{\left(2\right)}=Tr\left({L}^{2}\right)=2{z}_{1}^{2}+2{z}_{2}^{2}+4{e}^{{z}_{3}-{z}_{4}}$$

Now let us focus on the integrability issues. We know that
$n$ integrals of motion are associated with each generator of non-Noether
symmetry and according to the Liouville-Arnold theorem Hamiltonian system is
completely integrable if it possesses $n$ functionally independent integrals of
motion in involution (two functions $f$ and $g$ are said to be
in involution if their Poisson bracket vanishes $\{f,g\}=0$).
Generally speaking the conservation laws associated with symmetry might appear to be neither
independent nor involutive.
However it is reasonable to ask the question – what condition should be satisfied
by the generator of the symmetry to ensure the involutivity
($\{{Y}^{\left(k\right)},{Y}^{\left(m\right)}\}=0$) of conserved quantities?
In Lax theory such a condition is known as
Classical Yang-Baxter Equation (CYBE). Since involutivity of the conservation laws
is closely related to the integrability it is essential to have some analog of CYBE for the generator
of non-Noether symmetry. To address this issue we would like to propose the following theorem.

If the vector field $E$ on $2n$-dimensional
Poisson manifold $M$satisfies the condition
$$\left[\right[E[E,W]\left]W\right]=0$$
and $W$ bivector field has maximal rank (${W}^{n}\ne 0$)
then the functions (10) are in involution
$$\{{Y}^{\left(k\right)},{Y}^{\left(m\right)}\}=0$$

First of all let us note that
the identity (1) satisfied by the Poisson
bivector field $W$ is responsible for the Liouville theorem
$$[W,W]=0\text{}{L}_{W\left(f\right)}W=\left[W\right(f),W]=0$$
By taking the Lie derivative of the expression (1)
we obtain another useful identity
$${L}_{E}[W,W]=\left[E\right[W,W\left]\right]=\left[\right[E,W\left]W\right]+\left[W\right[E,W\left]\right]=2[\u0174,W]=0.$$
This identity gives rise to the following relation
$$[\u0174,W]=0\text{}\iff \text{}\left[\u0174\right(f),W]=-[\u0174,W(f\left)\right]$$
and finally condition (39) ensures third identity
$$[\u0174,\u0174]=0$$
yielding Liouville theorem for $\u0174$
$$[\u0174,\u0174]=0\text{}\iff \text{}\left[\u0174\right(f),\u0174]=0$$
Indeed
$$[\u0174,\u0174]=\left[\right[E,W\left]\u0174\right]=\left[\right[\u0174,E\left]W\right]\phantom{\rule{0ex}{0ex}}=-\left[\right[E,\u0174\left]W\right]=-\left[\right[E[E,W]\left]W\right]=0$$
Now let us consider two different solutions ${c}_{i}\ne {c}_{j}$
of the equation (18). By taking the Lie derivative of the equation
$$(\u0174-{c}_{i}W{)}^{n}=0$$
along the vector fields $W\left({c}_{j}\right)$ and
$\u0174\left({c}_{j}\right)$ and using Liouville theorem for
$W$ and $\u0174$ bivectors we obtain the following relations
$$(\u0174-{c}_{i}W{)}^{n-1}({L}_{W\left({c}_{j}\right)}\u0174-\{{c}_{j},{c}_{i}\}W)=0,$$
and
$$(\u0174-{c}_{i}W{)}^{n-1}({c}_{i}{L}_{\u0174\left({c}_{j}\right)}W+\{{c}_{j},{c}_{i}{\}}_{\text{\u2022}}W)=0,$$
where
$$\{{c}_{i},{c}_{j}{\}}_{\text{\u2022}}=\u0174(d{c}_{i}\wedge d{c}_{j})$$
is the Poisson bracket calculated by means of the bivector field $\u0174$.
Now multiplying (48) by ${c}_{i}$ subtracting (49) and using
identity (43) gives rise to
$$\left(\right\{{c}_{i},{c}_{j}{\}}_{\text{\u2022}}-{c}_{i}\{{c}_{i},{c}_{j}\}\left)\right(\u0174-{c}_{i}W{)}^{n-1}W=0$$
Thus, either
$$\{{c}_{i},{c}_{j}{\}}_{\text{\u2022}}-{c}_{i}\{{c}_{i},{c}_{j}\}=0$$
or the volume field
$(\u0174-{c}_{i}W{)}^{n-1}W$
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 ${W}^{n}=0$ 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
$$\{{c}_{i},{c}_{j}{\}}_{\text{\u2022}}-{c}_{j}\{{c}_{i},{c}_{j}\}=0$$
Finally by comparing (52) and (53) we obtain that
the functions ${c}_{i}$ are in involution with respect to the both
Poisson structures (since ${c}_{i}\ne {c}_{j}$)
$$\{{c}_{i},{c}_{j}{\}}_{\text{\u2022}}=\{{c}_{i},{c}_{j}\}=0$$
and according to (19) the same is true for the integrals of motion
${Y}^{\left(k\right)}$.

Each generator of non-Noether symmetry satisfying equation (39) endows
dynamical system with the bi-Hamiltonian structure – couple ($W,\u0174$)
of compatible ($[W,\u0174]=0$)
Poisson ($[W,W]=[\u0174,\u0174]=0$)
bivector fields.

Theorem 3 is useful in multidimentional dynamical systems where involutivity of
conservation laws can not be checked directly.

One can check that the non-Noether symmetry (23) satisfies
condition (39) and the bivector fields $W$ and $\u0174$ defined by
(20) and (24) form bi-Hamiltonian system
$[W,W]=[W,\u0174]=[\u0174,\u0174]=0$.

Another concept that is often used in theory of dynamical systems and could
be related to the non-Noether symmetry is the bidifferential calculus (bicomplex approach).
Recently A. Dimakis and F. Müller-Hoissen
applied bidifferential calculi to the wide range of integrable models
including KdV hierarchy, KP equation, self-dual Yang-Mills equation,
Sine-Gordon equation, Toda models, non-linear Schrödinger
and Liouville equations. It turns out that these models can be effectively
described and analyzed using the bidifferential calculi [1], [2].

Under the bidifferential calculus we mean the graded algebra of differential forms
$$\Omega =\underset{k=0}{\overset{\mathrm{\infty}}{\cup}}{\Omega}^{\left(k\right)}$$
(${\Omega}^{\left(k\right)}$ denotes the space of $k$-degree differential forms)
equipped with a couple of differential operators
$$d,\text{\u0111}:{\Omega}^{\left(k\right)}\to {\Omega}^{(k+1)}$$
satisfying
${d}^{2}={\text{\u0111}}^{2}=d\text{\u0111}+\text{\u0111}d=0$
conditions (see [2]).
It is interesting that if generator of the non-Noether symmetry satisfies
equation (39) then we are able to construct an invariant bidifferential calculus
of a certain type. This construction is summarized in the following theorem:

Let $(M,h)$ be regular Hamiltonian system on the Poisson manifold $M$.
Then, if the vector field $E$ on $M$ generates the non-Noether symmetry
and satisfies the equation (39), the differential operators
$$du={\Phi}^{-1}\left(\right[W,\Phi \left(u\right)\left]\right)$$
$$\text{\u0111}u={\Phi}^{-1}\left(\right[[E,W]\Phi \left(u\right)\left]\right)$$
form invariant bidifferential calculus
(${d}^{2}={\text{\u0111}}^{2}=d\text{\u0111}+\text{\u0111}d=0$)
over the graded algebra of differential forms on $M$.

First of all we have to show that $d$ and $\text{\u0111}$
are really differential operators , i.e., they are linear maps from
${\Omega}^{\left(k\right)}$ into
${\Omega}^{(k+1)}$, satisfy derivation property and
are nilpotent (${d}^{2}={\text{\u0111}}^{2}=0$).
Linearity is obvious and follows from the linearity of the Schouten bracket $[,]$
and $\Phi ,{\Phi}^{-1}$
maps. Then, if $u$ is a $k$-degree form
$\Phi $ maps it on $k$-degree multivector field and
the Schouten brackets $[W,\Phi (u\left)\right]$ and
$\left[\right[E,W\left]\Phi \right(u\left)\right]$ result the
$k+1$-degree multivector fields that are mapped on $k+1$-degree
differential forms by ${\Phi}^{-1}$.
So, $d$ and $\text{\u0111}$
are linear maps from ${\Omega}^{\left(k\right)}$ into
${\Omega}^{(k+1)}$.
Derivation property follows from the same feature of the Schouten bracket
$[,]$ and linearity of
$\Phi $ and
${\Phi}^{-1}$ maps.
Now we have to prove the nilpotency of $d$ and $\text{\u0111}$.
Let us consider ${d}^{2}u$
$${d}^{2}u={\Phi}^{-1}\left(\right[W,\Phi \left({\Phi}^{-1}\right([W,\Phi (u\left)\right]\left)\right)\left]\right)\phantom{\rule{0ex}{0ex}}={\Phi}^{-1}\left(\right[W[W,\Phi (u\left)\right]\left]\right)=0$$
as a result of the property (41) and the Jacobi identity for $[,]$ bracket.
In the same manner
$${\text{\u0111}}^{2}u={\Phi}^{-1}\left(\right[[W,E]\left[\right[W,E\left]\Phi \right(u\left)\right]\left]\right)=0$$
according to the property (45) of
$[W,E]=\u0174$ and the Jacobi identity.
Thus, we have proved that $d$ and $\text{\u0111}$ are differential operators
(in fact $d$ is ordinary exterior differential and the expression
(57) is its well known representation in terms of Poisson bivector field).
It remains to show that the compatibility condition $d\text{\u0111}+\text{\u0111}d=0$
is fulfilled. Using definitions of $d,\text{\u0111}$ and the Jacobi identity we get
$$(d\text{\u0111}+\text{\u0111}d)\left(u\right)={\Phi}^{-1}\left(\right[\left[\right[W,E\left]W\right]\Phi \left(u\right)\left]\right)=0$$
as far as (43) is satisfied.
So, $d$ and $\text{\u0111}$ form the bidifferential calculus over the graded
algebra of differential forms.
It is also clear that the bidifferential calculus $d,\text{\u0111}$
is invariant, since both $d$ and $\text{\u0111}$ commute with time evolution
operator $W\left(h\right)=\{h,\}$.

Conservation laws that are associated with the bidifferential calculus
(57) (58)
and form Lenard scheme (see [2]):
$$(k+1)\text{\u0111}{I}^{\left(k\right)}=kd{I}^{(k+1)}$$
coincide with the sequence of integrals of motion (36).
Proof of this correspondence lay outside the scope of present article,
but could be done in the manner similar to [1].

The symmetry (23) endows
${R}^{4}$ with bicomplex structure $d,\text{\u0111}$ where $d$ is ordinary
exterier derivative while $\text{\u0111}$ is defined by
$$\text{\u0111}{z}_{1}={z}_{1}d{z}_{1}-{e}^{{z}_{3}-{z}_{4}}d{z}_{4}\phantom{\rule{0ex}{0ex}}\text{\u0111}{z}_{2}={z}_{2}d{z}_{2}+{e}^{{z}_{3}-{z}_{4}}d{z}_{3}\phantom{\rule{0ex}{0ex}}\text{\u0111}{z}_{3}={z}_{1}d{z}_{3}+d{z}_{2}\phantom{\rule{0ex}{0ex}}\text{\u0111}{z}_{4}={z}_{2}d{z}_{4}-d{z}_{1}$$
and is extended to whole De Rham complex by linearity, derivation property and
compatibility property $d\text{\u0111}+\text{\u0111}d=0$. The conservation laws
${I}^{\left(1\right)}$ and ${I}^{\left(2\right)}$ defined by (38)
form the simpliest Lenard scheme
$$2\text{\u0111}{I}^{\left(1\right)}=d{I}^{\left(2\right)}$$

Finally we would like to reveal some features of the operator
${\text{\u0154}}_{E}$
(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
$T\left(F\right)$ vanishes, i. e. whenever for arbitrary vector fields $X,Y$ the condition
$$T\left(F\right)(X,Y)=[FX,FY]-F\left(\right[FX,Y]+[X,FY]-F[X,Y\left]\right)=0$$
is satisfied.
Torsionless forms are also called Frölicher-Nijenhuis operators and are widely used in
theory of integrable models. We would like to show
that each generator of non-Noether symmetry satisfying equation (39)
canonnically leads to invariant Frölicher-Nijenhuis operator on tangent
bundle over the phase space. Strictly speaking we have the following theorem.

Let $(M,h)$ be regular Hamiltonian system on the Poisson manifold $M$.
If the vector field $E$ on $M$ generates the non-Noether symmetry
and satisfies the equation (39) then the linear operator, defined for
every vector field $X$ by equation
$${R}_{E}\left(X\right)=\Phi \left({L}_{E}{\Phi}^{-1}\right(X\left)\right)-[E,X]$$
is invariant Frölicher-Nijenhuis operator on $M$.

Invariance of ${R}_{E}$ follows from the invariance of the
${\text{\u0154}}_{E}$ defined by (31)
(note that for arbitrary 1-form vector field $u$ and vector field $X$
contraction ${i}_{X}u$ has the property
${i}_{{R}_{E}X}u={i}_{X}{\text{\u0154}}_{E}u$,
so ${R}_{E}$ is actually transposed to ${\text{\u0154}}_{E}$).
It remains to show that the condition (39) ensures vanishing of the
Frölicher-Nijenhuis torsion $T\left({R}_{E}\right)$ of
${R}_{E}$, i.e. for arbitrary vector fields $X,Y$ we must get
$$T\left({R}_{E}\right)(X,Y)=\left[{R}_{E}\right(X),{R}_{E}(Y\left)\right]\phantom{\rule{0ex}{0ex}}-{R}_{E}\left(\right[{R}_{E}\left(X\right),Y]+[X,{R}_{E}\left(Y\right)]-{R}_{E}([X,Y]\left)\right)=0$$
First let us introduce the following auxiliary 2-forms
$$\omega ={\Phi}^{-1}\left(W\right),\text{}{\omega}^{\text{\u2022}}={\text{\u0154}}_{E}\omega \text{}{\omega}^{\text{\u2022}\text{\u2022}}={\text{\u0154}}_{E}{\omega}^{\text{\u2022}}$$
Using the realization (57) of the differential $d$
and the property (1) yields
$$d\omega ={\Phi}^{-1}\left(\right[W,W\left]\right)=0$$
Similarly, using the property (43) we obtain
$$d{\omega}^{\text{\u2022}}=d{\Phi}^{-1}\left(\right[E,W\left]\right)-d{L}_{E}\omega ={\Phi}^{-1}\left(\right[[E,W]W\left]\right)-{L}_{E}d\omega =0$$
And finally, taking into account that
${\omega}^{\text{\u2022}}=2{\Phi}^{-1}\left(\right[E,W\left]\right)$
and using the condition (39), we get
$$d{\omega}^{\text{\u2022}\text{\u2022}}=2{\Phi}^{-1}\left(\right[\left[E\right[E,W\left]\right]W\left]\right)-2d{L}_{E}{\omega}^{\text{\u2022}}=-2{L}_{E}d{\omega}^{\text{\u2022}}=0$$
So the differential forms
$\omega ,{\omega}^{\text{\u2022}},{\omega}^{\text{\u2022}\text{\u2022}}$
are closed
$$d\omega =d{\omega}^{\text{\u2022}}=d{\omega}^{\text{\u2022}\text{\u2022}}=0$$
Now let us consider the contraction of $T\left({R}_{E}\right)$ and $\omega $.
$${i}_{T\left({R}_{E}\right)(X,Y)}\omega ={i}_{[{R}_{E}X,{R}_{E}Y]}\omega -{i}_{[{R}_{E}X,Y]}{\omega}^{\text{\u2022}}-{i}_{[X,{R}_{E}Y]}{\omega}^{\text{\u2022}}+{i}_{[X,Y]}{\omega}^{\text{\u2022}\text{\u2022}}\phantom{\rule{0ex}{0ex}}={L}_{{R}_{E}X}{i}_{Y}{\omega}^{\text{\u2022}}-{i}_{{R}_{E}Y}{L}_{X}{\omega}^{\text{\u2022}}-{L}_{{R}_{E}X}{i}_{Y}{\omega}^{\text{\u2022}}+{i}_{Y}{L}_{{R}_{E}X}{\omega}^{\text{\u2022}}-{L}_{X}{i}_{{R}_{E}Y}{\omega}^{\text{\u2022}}\phantom{\rule{0ex}{0ex}}+{i}_{{R}_{E}Y}{L}_{X}{\omega}^{\text{\u2022}}+{i}_{[X,Y]}{\omega}^{\text{\u2022}\text{\u2022}}={i}_{Y}{L}_{X}{\omega}^{\text{\u2022}\text{\u2022}}-{L}_{X}{i}_{Y}{\omega}^{\text{\u2022}\text{\u2022}}+{i}_{[X,Y]}{\omega}^{\text{\u2022}\text{\u2022}}=0$$
where we used (68) (72),
the property of the Lie derivative
$${L}_{X}{i}_{Y}\omega ={i}_{Y}{L}_{X}\omega +{i}_{[X,Y]}\omega $$
and the relations of the following type
$${L}_{{R}_{E}X}\omega =d{i}_{{R}_{E}X}\omega +{i}_{{R}_{E}X}d\omega =d{i}_{X}{\omega}^{\text{\u2022}}={L}_{X}{\omega}^{\text{\u2022}}-{i}_{X}d{\omega}^{\text{\u2022}}={L}_{X}{\omega}^{\text{\u2022}}$$
So we proved that for arbitrary vector fields $X,Y$
the contraction of $T\left({R}_{E}\right)(X,Y)$ and $\omega $ vanishes.
But since $W$ bivector is non-degenerate
(${W}^{n}\ne 0$), its counter image
$$\omega ={\Phi}^{-1}\left(W\right)$$
is also non-degenerate and vanishing of the contraction (73)
implies that the torsion $T\left({R}_{E}\right)$ itself is zero.
So we get
$$T\left({R}_{E}\right)(X,Y)=\left[{R}_{E}\right(X),{R}_{E}(Y\left)\right]\phantom{\rule{0ex}{0ex}}-{R}_{E}\left(\right[{R}_{E}\left(X\right),Y]+[X,{R}_{E}\left(Y\right)]-{R}_{E}([X,Y]\left)\right)=0$$

Note that operator ${R}_{E}$ associated with non-Noether
symmetry (23) reproduces well known Frölicher-Nijenhuis operator
$${R}_{E}={z}_{1}d{z}_{1}\otimes {D}_{1}-d{z}_{1}\otimes {D}_{4}+{z}_{2}d{z}_{2}\otimes {D}_{2}+d{z}_{2}\otimes {D}_{3}+{z}_{1}d{z}_{3}\otimes {D}_{3}\phantom{\rule{0ex}{0ex}}+{e}^{{z}_{3}-{z}_{4}}d{z}_{3}\otimes {D}_{2}+{z}_{2}d{z}_{4}\otimes {D}_{4}-{e}^{{z}_{3}-{z}_{4}}d{z}_{4}\otimes {D}_{1}$$
(compare with [3])

In summary let us note that the non-Noether symmetries form quite interesting
class of symmetries of Hamiltonian dynamical system and lead not only to
a number of conservation laws (that under certain conditions ensure integrability),
but also enrich the geometry of the phase space by endowing it with several important
structures, such as Lax pair, bicomplex,
bi-Hamiltonian structure, Frölicher-Nijenhuis operators etc.
The present paper attempts to emphasize deep relationship between different
concepts used in construction of conservation laws and non-Noether symmetry.

Author is grateful to Zakaria Giunashvili, George Jorjadze and
Michael Maziashvili for constructive discussions and help.
This work was supported by INTAS (00-00561).

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