We discuss geometric properties of non-Noether symmetries and their possible applications in integrable Hamiltonian systems. Correspondence between non-Noether symmetries and conservation laws is revisited. It is shown that in regular Hamiltonian systems such symmetries canonically lead to Lax pairs on the algebra of linear operators on cotangent bundle over the phase space. Relationship between non-Noether symmetries and other widespread 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 a continuous non-Noether symmetry are in involution whenever the generator of the symmetry satisfies a certain Yang-Baxter type equation. Action of one-parameter group of symmetry on algebra of integrals of motion is studied and involutivity of group orbits is discussed. Hidden non-Noether symmetries of Toda chain, Korteweg-de Vries equation, Benney system, nonlinear water wave equations and Broer-Kaup system are revealed and discussed.

Symmetries play essential role in dynamical systems, because they usually simplify analysis of evolution equations and often provide quite elegant solution of problems that otherwise would be difficult to handle. In Lagrangian and Hamiltonian dynamical systems special role is played by Noether symmetries — an important class of symmetries that leave action invariant and have some exceptional features. In particular, Noether symmetries deserved special attention due to celebrated Noether's theorem, that established correspondence between symmetries, that leave action functional invariant, and conservation laws of Euler-Lagrange equations. This correspondence can be extended to Hamiltonian systems where it becomes more tight and evident then in Lagrangian case and gives rise to Lie algebra homomorphism between Lie algebra of Noether symmetries and algebra of conservation laws (that form Lie algebra under Poisson bracket).

Role of symmetries that are not of Noether type has been suppressed for quite a long time.
However, after some publications of Hojman, Harleston, Lutzky and others
(see

Existence of correspondence between non-Noether symmetries and conserved quantities
raised many questions concerning relationship among this type of symmetries and
other geometric structures emerging in theory of integrable models.
In particular one could notice suspicious similarity between the method of constructing
conservation laws from generator of non-Noether symmetry and
the way conserved quantities are produced in either Lax theory, bi-Hamiltonian formalism,
bicomplex approach or Lenard scheme.
It also raised natural question whether set of conservation laws associated with non-Noether
symmetry is involutive or not, and since it appeared that in general it may not be involutive,
there emerged the need of involutivity criteria, similar to Yang-Baxter equation used in Lax theory
or compatibility condition in bi-Hamiltonian formalism and bicomplex approach.
It was also unclear how to construct conservation laws in case of infinite dimensional
dynamical systems where volume forms used in Lutzky's construction are no longer well defined.
Some of these questions were addressed in papers

Review is organized as follows. In first section we briefly recall some aspects of geometric
formulation of Hamiltonian dynamics. Further, in second section, correspondence
between non-Noether symmetries and integrals of motion in regular Hamiltonian systems is
discussed. Lutzky's theorem is reformulated in terms of bivector fields
and alternative derivation of conserved quantities suitable for computations in infinite
dimensional Hamiltonian dynamical systems is suggested. Non-Noether symmetries of
two and three particle Toda chains are used to illustrate general theory.
In the subsequent section geometric formulation of Hojman's theorem

Subsequent sections of present review provide examples of integrable models
that possess interesting non-Noether symmetries. In particular section 10 reveals
non-Noether symmetry of

The basic concept in geometric formulation of Hamiltonian dynamics is notion of symplectic manifold. Such a manifold plays the role of the phase space of the dynamical system and therefore many properties of the dynamical system can be quite effectively investigated in the framework of symplectic geometry. Before we consider symmetries of the Hamiltonian dynamical systems, let us briefly recall some basic notions from symplectic geometry.

The symplectic manifold is a pair ^{n} ≠ 0

The symplectic form ^{*}M^{*}M

One of the nice features of locally Hamiltonian vector fields, known as Liouville's theorem,
is that these vector fields preserve symplectic form

Clearly, Hamiltonian vector fields constitute subset of locally Hamiltonian ones since
every exact 1-form is also closed. Moreover one can notice that Hamiltonian vector fields form
ideal in algebra of locally Hamiltonian vector fields. This fact can be observed as follows.
First of all for arbitrary couple of locally Hamiltonian vector fields

Isomorphism ^{n} ≠ 0^{p + q}[C

Being counter image of symplectic form,

In Hamiltonian dynamical systems Poisson bivector field is geometric object that
underlies definition of Poisson bracket — kind of Lie bracket on algebra of
smooth real functions on phase space. In terms of bivector field

To define dynamics on

Now let us focus on symmetries of Hamilton's equation

Before we proceed
let us recall that each vector field ^{zLE}
^{zLE}(f) =
f + zL^{2}f + ⋯

Further one should distinguish between groups of symmetry transformations generated by Hamiltonian,
locally Hamiltonian and non-Hamiltonian vector fields. First kind of symmetries
are known as Noether symmetries and are widely used in Hamiltonian dynamics due to their
tight connection with conservation laws. Second group of symmetries is rarely used.
While third group of symmetries that further will be referred
as non-Noether symmetries seems to play important role in integrability issues due to
their remarkable relationship with bi-Hamiltonian structures and
Frölicher-Nijenhuis operators. Thus if in addition to

Now let us focus on non-Noether symmetries. We would like to show that the presence of
such a symmetry essentially enriches the geometry of the phase space
and under the certain conditions can 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. More precisely the
relationship between non-Noether symmetries and the conservation laws is described by
the following theorem. This theorem was proposed by Lutzky in

^{(k)} = ^{k} ∧ W^{n − k}^{n}

^{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}
+ (n − k)[W(h) , W] ∧ Ŵ^{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)} =
^{(k)}^{(k)} = ^{k} ∧ ω^{n − k}^{n}^{n} = 0
^{n}
^{(k)} = i^{k}^{k}
^{k}^{k}^{(k)}^{(k)}^{(k)} =
^{(k)}

Besides continuous non-Noether symmetries generated by non-Hamiltonian
vector fields one may encounter discrete non-Noether symmetries — noncannonical
transformations that doesn't necessarily form group but commute with evolution operator
^{(k)} = ^{k} ∧ W^{n − k}^{n}

^{4}^{2} +
^{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} ^{k} ∧ W^{n − k}^{z3 − z4})
^{(1)} = ^{(2)} = ^{z3 − z4}
^{6}^{2} +
^{2} +
^{2} +
e^{z4 − z5} +
e^{z5 − z6}
^{2} − 2e^{z4 − z5} −
^{z4 − z5}^{2} + 3e^{z4 − z5} −
e^{z5 − z6} +
^{z4 − z5}^{2} + 2e^{z5 − z6} +
^{z5 − z6}
^{2} + e^{z4 − z5})^{2} + e^{z4 − z5} +
e^{z5 − z6})^{2} + e^{z5 − z6})
^{z4 − z5} ^{z5 − z6} ^{k} ∧ W^{n − k}^{4}^{(1)} = ^{(2)} = ^{z4 − z5} − e^{z5 − z6})
= ^{(3)} = z^{z4 − z5} −
z^{z5 − z6}
=

Besides Hamiltonian dynamical systems that admit invariant symplectic form
^{n}

In fact theorem is valid for larger class of symmetries. Namely one can consider
symmetries with time dependent generators. Note however that in this case condition
^{(m)} = (L^{m}Ω

^{z4 − z5}^{z4 − z5} − e^{z5 − z6}^{z5 − z6}
^{(1)} = L^{2} +
^{2} +
^{2} +
e^{z4 − z5} +
e^{z5 − z6}^{(2)} = L^{(1)} =
^{3} + z^{3} + z^{3}) ^{z4 − z5} +
^{z5 − z6}

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 that plays quite important role in construction
of completely integrable models.
Let us recall that Lax pair of Hamiltonian system on Poisson manifold ^{(k)} =
^{k})
^{(k)} = ^{k}) =
^{k}^{k − 1}^{k − 1}[L , P]) = ^{k}, P]) = 0

^{2}h

^{2}h^{2}h

The conservation laws ^{(k)} = ^{k}) = ^{k}
^{(k)}^{(k)}^{(m)} + (− 1)^{m}mC^{(m)} +
^{k} I^{(m − k)}C^{(k)} = 0

^{z3 − z4}^{z3 − z4}^{z3 − z4}^{z3 − z4}^{z3 − z4}^{z3 − z4}^{(1)} = ^{(2)} = ^{2}) =
z^{2} + z^{2} + 2e^{z3 − z4}
^{z4 − z5}^{z4 − z5}^{z5 − z6}^{z5 − z6}^{z4 − z5}^{z4 − z5}^{z4 − z5}^{z4 − z5} − e^{z5 − z6}^{z5 − z6}^{z5 − z6}^{z5 − z6}^{(1)} = ^{(2)} = ^{2}) =
z^{2} + z^{2} + z^{2} +
2e^{z4 − z5} + 2e^{z5 − z6}^{(3)} = ^{3}) =
z^{3} + z^{3} + z^{3} +
3(z^{z4 − z5} +
3(z^{z5 − z6}

Now let us focus on the integrability issues. We know that

Now let us look at conservation laws ^{(1)}, Y^{(2)} ... Y^{(n)}^{(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 4 is useful in multidimensional dynamical systems where involutivity of conservation laws can not be checked directly.

Further we will focus on non-Noether symmetries that satisfy condition

Originally bi-Hamiltonian structures were introduced by F. Magri in analisys of integrable infinite dimensional Hamiltonian systems such as Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) hierarchies, Nonlinear Schrödinger equation and Harry Dym equation. Since that time bi-Hamiltonian formalism is effectively used in construction of involutive families of conservation laws in integrable models

Generic bi-Hamiltonian structure on ^{n} ≠ 0^{n} ≠ 0

Bi-Hamiltonian systems obtained by taking Lie derivative of Poisson bivector
along some vector field were studied in

^{z3 − z4} ^{z4 − z5} ^{z5 − z6}

In terms of differential forms bi-Hamiltonian structure is formed by couple of
closed differential 2-forms: symplectic form ^{n} ≠ 0^{∗} = L^{∗} = dL^{∗} + dL^{∗} + dh^{∗} = 0
^{∗} = L^{∗}^{∗}^{∗} = 0, ω^{n} ≠ 0
^{∗} + dh^{∗} = 0
^{∗} = L

The answer depends on ^{∗}^{∗}^{∗}^{∗} = dθ^{∗}^{∗}^{∗}^{∗}
^{∗}^{∗}
^{∗}^{∗}^{∗}^{∗} = ω^{∗}
^{∗}, X^{∗}^{∗}^{∗}(h)
− h^{∗}) = − dh'
^{∗}]^{∗}^{∗}^{∗} − X^{∗}^{∗}^{∗}
^{∗}^{∗} = L

Another important concept that is often used in theory of dynamical systems and may
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

Before we proceed let us specify what kind of bidifferential calculi we plan to consider.
Under the bidifferential calculus we mean the graded algebra of differential forms
over the phase space
^{(k)}
^{(k)}^{(k)} → Ω^{(k + 1)}
^{2} = đ^{2} = dđ + đd = 0

^{2} = đ^{2} = dđ + đd = 0

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

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

^{4}^{z3 − z4}dz^{z3 − z4}dz^{2} = 0^{2}z^{2}z^{z3 − z4}dz^{z3 − z4}đz^{z3 − z4}đz^{z3 − z4}đdz^{z3 − z4}đz^{z3 − z4}đz^{z3 − z4}dđz^{z3 − z4}dz^{z3 − z4}dz^{z3 − z4}đz^{z3 − z4}dz^{z3 − z4}dz^{z3 − z4}đz^{z3 − z4}dz^{z3 − z4}dđz^{z3 − z4}dz^{2}z^{2}z^{2}z^{2} = 0^{(1)} = z^{(2)} = z^{2} + z^{2}
+ 2e^{z3 − z4}
^{(1)} = dI^{(2)}
^{z4 − z5}dz^{z4 − z5}dz^{z5 − z6}dz^{z5 − z6}dz^{(1)} = z^{(2)} =
z^{2} + z^{2} + z^{2} +
2e^{z4 − z5} + 2e^{z5 − z6}^{(3)} =
z^{3} + z^{3} + z^{3} +
3(z^{z4 − z5} +
3(z^{z5 − z6}
^{(1)} = dI^{(2)}
^{(2)} = 2dI^{(3)}

Finally we would like to reveal some features of the operator

^{∗} = Ŕ^{∗∗} = Ŕ^{∗}
^{∗} =
dΦ^{∗} = 2Φ^{∗∗} =
2Φ^{∗} =
− 2L^{∗} = 0
^{∗}, ω^{∗∗}^{∗} = dω^{∗∗} = 0
^{∗} −
i^{∗} +
i^{∗∗}^{∗} −
i^{∗} −
L^{∗} +
i^{∗} −
L^{∗} +
i^{∗} +
i^{∗∗}^{∗∗} −
L^{∗∗} +
i^{∗∗} = 0
^{∗}^{∗} −
i^{∗} = L^{∗}
^{n} ≠ 0

^{z3 − z4}dz^{z3 − z4}dz^{(1)} = z^{(2)} = z^{2} + z^{2}
+ 2e^{z3 − z4}
^{(1)}) = dI^{(2)}
^{z4 − z5}dz^{z4 − z5}dz^{z5 − z6}dz^{z5 − z6}dz^{(1)} = z^{(2)} =
z^{2} + z^{2} + z^{2} +
2e^{z4 − z5} + 2e^{z5 − z6}^{(3)} =
z^{3} + z^{3} + z^{3}^{z4 − z5}
+ 3(z^{z5 − z6}
^{(3)} = 3Ŕ^{(2)}) =
6(Ŕ^{2}(dI^{(1)})

One-parameter group of transformations ^{zLE}J =
J + zL^{2}J + ...

Such an orbit ^{zLE}

^{(m)} = (L^{m}J {J^{(m)}, J^{(k)}} = 0

^{2}(W) = R([W(s),W]) = ½([W(s)[W(s),W]] − Φ^{2}ω))^{2}ω) =
Φ^{2})(W)
^{2}) = 2R^{2}
^{2}
^{m}(W)(L^{m + 1}(W)(dJ)
^{m}dJ) = c^{m}(W)(dJ)
^{m}dJ) + W((L^{m + 1}dJ) ^{m + 1}(W)(dJ) + c^{m + 1}(W)(dJ)
^{k}W((L^{m}dJ) = c^{k + m}(W)(dJ)
^{m + 1}dJ)
= c^{m + 1}(W)(dJ)
^{m − 1} = c^{m − 1}
^{m}J^{m}J, (L^{k}J} =
W(d(L^{m}J ∧ d(L^{k}J) ^{m}dJ ∧ (L^{k}dJ) =
c

Further we will use this theorem to prove involutivity of family of conservation laws constructed using non-Noether symmetry of Toda chain.

To illustrate features of non-Noether symmetries we often
refer to two and three particle non-periodic Toda systems.
However it turns out that non-Noether symmetries are present in
generic n-particle non-periodic Toda chains as well, moreover they preserve
basic features of symmetries

First of all let us remind that Toda model is
^{qs − 1 − qs} −
ε(n − s)e^{qs − qs + 1}
^{2} +
^{qs − qs + 1}
^{2})^{2})
^{qs − 1 − qs}
(E(q^{qs − qs + 1}
(E(q^{2} +
ε(s − 1)(n − s + 2)e^{qs − 1 − qs} −
ε(n − s)(n − s) e^{qs − qs + 1}^{qs − 1 − qs} −
ε(n − s)(p^{qs − qs + 1})^{2} +
ε(s − 1)e^{qs − 1 − qs} +
ε(n − s)e^{qs − qs + 1})
^{qs − qs + 1} ^{(1)} = C^{(1)} = ^{(2)} = (C^{(1)})^{2} − 2C^{(2)} =
^{2} +
2^{qs − qs + 1}^{(3)} = C^{(1)})^{3} − 3C^{(1)}C^{(2)}
+ 3C^{(3)} = ^{3} +
3^{qs − qs + 1}^{(4)} = C^{(1)})^{4} − 4(C^{(1)})^{2}C^{(2)} +
2(C^{(2)})^{2} + 4C^{(1)}C^{(3)} − 4C^{(4)}^{4} +
4^{2} + 2p^{2})
e^{qs − qs + 1}^{2(qs − qs + 1)} +
4^{qs − qs + 2} ^{(m)} = (− 1)^{m + 1}mC^{(m)} +
^{k + 1}I^{(m − k)}C^{(k)}
^{(k)} , C^{(m)}} = 0

Using formula ^{qk − qk + 1}^{qk − 1 − qk}
− ε(n − k)e^{qk − qk + 1}^{qk − qk + 1}^{qk − 1 − qk}

Like two and three particle Toda chain, n-particle Toda model also admits
invariant bidifferential calculus on algebra of differential forms over the phase space.
This bidifferential calculus can be constructed using non-Noether symmetry (see ^{qs − qs + 1}dq^{qs − 1 − qs}dq^{2} = 0^{(k)} = kdI^{(k + 1)}

Further let us focus on Frölicher-Nijenhuis geometry. Using formula ^{qs − qs + 1}dq^{qs − 1 − qs}dq^{(k)}^{(k)}) = kdI^{(k + 1)}

Finally, in order to outline possible applications of Theorem 8 let us study
action of non-Noether symmetry ^{(1)} + ^{2}^{(2)} +
^{3}^{(3)} + ⋯
^{(m)} = (L^{m}J
^{aLE}^{(1)} = L^{2} + ^{qs − qs + 1}^{(2)} = L^{(1)} = (L^{2}J =
^{3} +
^{qs − qs + 1}^{(3)} = L^{(2)} = (L^{3}J =
¾ ^{4} +
3^{2} + 2p^{2})e^{qs − qs + 1}^{2(qs − qs + 1)} +
3^{qs − qs + 2}^{(m)} = L^{(m − 1)} = (L^{m}J

Involutivity of this set of conservation laws can be verified using Theorem 8.
In particular one can notice that differential 1-form

Toda model provided good example of finite dimensional integrable Hamiltonian system
that possesses non-Noether symmetry. However there are many
infinite dimensional integrable Hamiltonian systems and in this case in
order to ensure integrability one should construct
infinite number of conservation laws. Fortunately in several integrable models
this task can be effectively simplified by identifying appropriate non-Noether symmetry.
First let us consider well known infinite dimensional integrable Hamiltonian system –
Korteweg-de Vries equation (KdV). The KdV equation has the following form
^{2}^{2})

Later we will focus on the symmetry generated by the following vector field
^{2} + ^{2}u

If ^{2} − ^{3}

Now let us show how non-Noether symmetry can be used to construct conservation laws
of KdV hierarchy. By integrating KdV it is easy to show that
^{(0)} = ^{(0)}^{(m)} = (L^{m}J^{(0)}^{(0)} = ^{(1)} = L^{(0)} =
¼^{2} dx ^{(2)} = (L^{2}J^{(0)} =
^{3}^{2}^{(3)} = (L^{3}J^{(0)} ^{4} −
^{2} + u^{2}^{(m)} = (L^{m}J^{(0)}

Among nonlinear partial differential equations that describe propagation of waves in shallow water
there are many remarkable integrable systems. We have already discussed case of KdV equation,
that possess non-Noether symmetries leading to the infinite sequence of conservation laws
and bi-Hamiltonian realization of these equations,
now let us consider other important water wave systems.
It is reasonable to start with dispersive water wave system ^{2})^{2})^{2})
^{2} − 2uv + uw^{2} + 4vw − 3v^{2}w − uu^{2} + 3vw^{2} − 3v^{2} + 2w^{2} + 6vw − w^{3} − 3ww

Before we proceed let us note that dispersive water wave system is actually infinite dimensional
Hamiltonian dynamical system. Assuming that ^{2}w + 2vw^{2} − 2v^{2})dx
^{∗} , u}^{∗} , v}^{∗} , w}^{∗} = − ¼ ^{2} + 2vw)dx

Now let us pay attention to conservation laws. By integrating third equation
of dispersive water wave system ^{(0)} =
^{(0)}^{(0)} = ^{(1)} = L^{(0)} =
− 2 ^{(2)} = L^{(1)} = (L^{2}J^{(0)} =
− 2^{2} + 2vw)dx^{(3)} = L^{(2)} = (L^{3}J^{(0)} =
− 6^{2}w + 2vw^{2} − 2v^{2})dx^{(4)} = L^{(3)} = (L^{4}J^{(0)}^{2}w^{2} + u^{2}w^{2}v − 6v^{2}w +
2vw^{3} − 3v^{2} − 2v^{(n)} = L^{(n − 1)} = (L^{n}J^{(0)}

Note that symmetry ^{2} − 2uv)^{2} + 4vw + x(uu^{2}w − 3v^{2} + 3vw^{2})^{2} − 4v + x(2ww^{2} + 6vw − w^{3})^{(0)} = ^{(1)} = L^{(0)} =
− 2 ^{(2)} = L^{(1)} = (L^{2}J^{(0)} =
− 2 ^{2} + 2vw)dx^{(3)} = L^{(2)} = (L^{3}J^{(0)} =
− 6 ^{2}w + 2vw^{2} − 2v^{2})dx^{(4)} = L^{(3)} = (L^{4}J^{(0)} =
− 24 ^{2}w^{2} − 2u^{2}v − 6v^{2}w + 2vw^{3})dx^{(n)} = L^{(n − 1)} = (L^{n}J^{(0)}

Another important integrable model that can be obtained from dispersive water wave system
is Broer-Kaup system ^{2} + 3vw^{2} + 3v^{2} − 2w^{3} − 3ww^{(0)} = ^{(1)} = L^{(0)} =
2 ^{(2)} = L^{(1)} = (L^{2}J^{(0)} =
4 ^{(3)} = L^{(2)} = (L^{3}J^{(0)} =
12 ^{2} + v^{2})dx^{(4)} = L^{(3)} = (L^{4}J^{(0)} =
24 ^{2}w + 2vw^{3} + 3v^{2} − 2v^{(n)} = L^{(n − 1)} = (L^{n}J^{(0)}

And exactly like in the dispersive water wave system one can rewrite equations of motion
^{2} + v^{2})dx
^{∗} , v}^{∗} , w}^{∗} = −
¼

By suppressing dispersive terms in Broer-Kaup system one reduces it to more simple
integarble model — dispersiveless long wave system ^{2} + vw^{2})^{2} + 4v + 2x(ww^{3})^{(0)} = ^{(1)} = L^{(0)} =
2 ^{(2)} = L^{(1)} = (L^{2}J^{(0)} =
4 ^{(3)} = L^{(2)} = (L^{3}J^{(0)} =
12 ^{2} + v^{2})dx^{(4)} = L^{(3)} = (L^{4}J^{(0)} =
48 ^{2}w + vw^{3})dx^{(n)} = L^{(n − 1)} = (L^{n}J^{(0)}

At the same time bi-Hamitonian structure of Broer-Kaup hierarchy, after reduction
gives rise to bi-Hamiltonian structure of dispersiveless long wave system

Among other reductions of dispersive water wave system one should probably mention
Burger's equation

Now let us consider another integrable system of nonlinear partial
differential equations — Benney system ^{2})^{2})^{2})
^{2} + x(2(uw)^{2}w + 3uw^{2})^{2} + vw^{2})^{2} + 4v + 2x(ww^{3} + 4vw + 4u)

At the same time, it is known fact, that under zero boundary conditions
^{2} + 4uv + v^{2}w)dx
^{∗} , u}^{∗} , v}^{∗} , w}^{∗} = ^{2} + 2uw)dx

The same symmetry yields infinite sequence of conservation laws of Benney system.
Namely one can construct sequence of integrals of motion by applying non-Noether
symmetry ^{(0)} = ^{(0)}^{(0)} = ^{(1)} = L^{(0)} =
2 ^{(2)} = L^{(1)} = (L^{2}J^{(0)} =
8 ^{(3)} = L^{(2)} = (L^{3}J^{(0)} =
12 ^{2} + 2uw)dx^{(4)} = L^{(3)} = (L^{4}J^{(0)} =
48 ^{2} + 4uv + v^{2}w)dx^{(5)} = L^{(4)} = (L^{5}J^{(0)} =
240 ^{2} + 8uvw + 2uw^{3} + 2v^{3} + v^{2}w^{2})dx^{(n)} = L^{(n − 1)} = (L^{n}J^{(0)}

The fact that many important integrable models, such as Korteweg-de Vries equation, Broer-Kaup system, Benney system and Toda chain, possess non-Noether symmetries that can be effectively used in analysis of these models, inclines us to think that non-Noether symmetries can play essential role in theory of integrable systems and properties of this class of symmetries should be investigated further. The present review indicates that in many cases non-Noether symmetries lead to maximal involutive families of functionally independent conserved quantities and in this way ensure integrability of dynamical system. To determine involutivity of conservation laws in cases when it can not be checked by direct computations (for instance one can not check directly the involutivity in many generic n-dimensional models like Toda chain and infinite dimensional models like KdV hierarchy) we propose analog of Yang-Baxter equation, that being satisfied by generator of symmetry, ensures involutivity of family of conserved quantities associated with this symmetry.

Another important feature of non-Noether symmetries is their relationship with several essential geometric concepts, emerging in theory of integrable systems, such as Frölicher-Nijenhuis operators, Lax pairs, bi-Hamiltonian structures and bicomplexes. On the one hand this relationship enlarges possible scope of applications of non-Noether symmetries in Hamiltonian dynamics and on the other hand it indicates that existence of invariant Frölicher-Nijenhuis operators, bi-Hamiltonian structures and bicomplexes in many cases can be considered as manifestation of hidden symmetries of dynamical system.