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.

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 ^{n} ≠ 0

Time evolution of observables (smooth functions on phase space) is governed by the Hamilton's equation
^{zLE}^{zLE}(Ô) = f + zL^{2}Ô + ⋯

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

^{(k)} = ^{k} ∧ W^{n − k}

^{k} ∧ W^{n − k} = Y^{(k)}W^{n}.
^{k} ∧ W^{n − k} =
^{(k)}^{n} + Y^{(k)}[W(h) , W^{n}]
^{k − 1} ∧ W^{n − k}^{k} ∧ W^{n − k − 1} = ^{(k)}^{n} + nY^{(k)}[W(h) , W] ∧ W^{n − 1}
^{(k)} W^{n} = 0
^{n} ≠ 0^{(k)}

Instead of conserved quantities
^{(1)} ... Y^{(n)}^{n} = 0
^{(k)}^{(k)} =

^{4}^{2} + ½z^{2} + e^{z3 − z4}
^{2} − e^{z3 − z4} −
^{z3 − z4}^{2} + 2e^{z3 − z4} +
^{z3 − z4}^{2} + e^{z3 − z4})^{2}
+ e^{z3 − z4})
^{z3 − z4}D^{k} ∧ W^{n − k}^{z3 − z4}) D^{(1)} = ^{(2)} = ^{z3 − z4}

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 ^{(k)} = Tr(L^{k})

^{−1})

^{− 1}([E , Φ(u)]) − L

The conservation laws ^{(k)} = Tr(L^{k}) = ^{k}

^{z3 − z4};
L^{(1)} = Tr(L) = 2(z^{(2)} = Tr(L^{2}) = 2z^{2} + 2z^{2}
+ 4e^{z3 − z4}

Now let us focus on the integrability issues. We know that
^{(k)} , Y^{(m)}} = 0

^{n} ≠ 0^{(k)} , Y^{(m)}} = 0

^{n} = 0
^{n − 1}(L^{n − 1}(c^{n − 1}W = 0
^{n − 1}W^{n − 1}W^{n} = 0^{(k)}

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

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

Under the bidifferential calculus we mean the graded algebra of differential forms
^{(k)}
^{(k)}^{(k)} → Ω^{(k + 1)}
^{2} = đ^{2} = dđ + đd = 0

^{− 1}([W , Φ(u)])
^{− 1}([[E , W]Φ(u)])
^{2} = đ^{2} = dđ + đd = 0

^{(k)}^{(k + 1)}^{2} = đ^{2} = 0^{− 1}^{− 1}^{(k)}^{(k + 1)}^{− 1}^{2}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

Conservation laws that are associated with the bidifferential calculus
^{(k)} = kdI^{(k + 1)}

^{4}^{z3 − z4}dz^{z3 − z4}dz^{(1)}^{(2)}^{(1)} = dI^{(2)}

Finally we would like to reveal some features of the operator

^{− 1}(X)) − [E , X]

^{− 1}(W), ω^{•} = Ŕ^{••} = Ŕ^{•}
^{− 1}([W , W]) = 0
^{•} = dΦ^{− 1}([E , W]) − dL^{− 1}([[E , W]W]) − L^{•} = 2Φ^{− 1}([E , W])^{••} = 2Φ^{− 1}([[E[E , W]]W]) − 2dL^{•}
= − 2L^{•} = 0
^{•}, ω^{••}^{•} = dω^{••} = 0
^{•} −
i^{•} +
i^{••}^{•} −
i^{•} −
L^{•} +
i^{•} −
L^{•}^{•} +
i^{••} =
i^{••} −
L^{••} +
i^{••} = 0
^{•} = L^{•} − i^{•}
= L^{•}
^{n} ≠ 0^{− 1}(W)

^{z3 − z4}dz^{z3 − z4}dz