In the present paper the non-Noether symmetries of the Toda model, nonlinear Schödinger equation and Korteweg-de Vries equations (KdV and mKdV) are discussed. It appears that these symmetries yield the complete sets of conservation laws in involution and lead to the bi-Hamiltonian realizations of the above mentioned models.

Because of their exceptional properties the non-Noether symmetries could be
effectively used in analysis of Hamiltonian dynamical systems.
From the geometric point of view these symmetries are important
because of their tight relationship with geometric structures on phase space
such as bi-Hamiltonian structures, Frölicher-Nijenhuis operators,
Lax pairs and bicomplexes

The n-particle non periodic Toda model is one of integrable models that possesses such a nontrivial symmetry. In this model non-Noether symmetry (which is one-parameter group of noncannonical transformations) yields conservation laws that appear to be functionally independent, involutive and ensure the integrability of this dynamical system. Well known bi-Hamiltonian realization of the Toda model is also related to this symmetry.

Nonlinear Schrödinger equation is another important example where symmetry (again one-parameter group) leads to the infinite sequence of conservation laws in involution. The KdV and mKdV equations also possess non-Noether symmetries which are quite nontrivial (but symmetry group is still one-parameter) and in each model the infinite set of conservation laws is associated with the single generator of the symmetry.

Before we consider these models in detail we briefly remind some basic facts
concerning symmetries of Hamiltonian systems. Since throughout the article
continuous one-parameter groups of symmetries play central role let us remind that
each vector field ^{zLE}
^{zLE}(f) =
f + zL^{2}f + ⋯
^{k}^{k} k = 1,2, ... n

Now let us focus on non-Noether symmetry of the Toda model –
^{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}
dq^{2} − Y^{2} +
^{qs − qs + 1}^{3}
− Y^{3} +
^{qs − qs + 1}^{4} −
Y^{2}Y^{2} +
Y^{4} +
^{2} + 2p^{2}) e^{qs − qs + 1}^{2(qs − qs + 1)} +
^{qs − qs + 2} ^{m}Y^{− 1}
^{k}I

Unlike the Toda model the dynamical systems in our next examples are
infinite dimensional and in order to ensure integrability one should construct
infinite number of conservation laws. Fortunately in several integrable models
this task could be effectively done by identifying appropriate non-Noether symmetry.
First let us consider well known infinite dimensional integrable Hamiltonian system –
nonlinear Schrödinger equation (NSE)
^{2}ū)
^{2}^{2}E(ū)
+ 4uūE(u)]
^{2})^{2}ū^{2}ū^{2} − u^{2} − 2Y^{3} − 3Y^{2}ū^{2} − u^{4} − 4Y^{2}Y^{2} + 4Y^{2}ū^{2}u^{m}mY^{k}I

Now let us consider other important integrable models –
Korteweg-de Vries equation (KdV) and modified Korteweg-de Vries equation (mKdV).
Here symmetries are more complicated but generator of the symmetry still can be
identified and used in construction of conservation laws. The KdV and mKdV equations
have the following form
^{2}u^{2}^{2}E(u)^{2})^{2} +
^{2}u^{3}
+ u^{2}u^{2}u^{3}
+ 30u^{4}u^{2}^{2} − ^{3}^{2} + u^{4}) dx [mKdV]
^{2} dx ^{3} − 3Y^{3}^{2}^{4} − 4Y^{2}Y^{2} + 4Y^{4} −
^{2} + u^{2}^{m}mY^{k}I^{2} dx ^{4} + u^{2}) dx ^{3} − 3Y^{6} + 10 u^{2}u^{2}
+ u^{2}) dx ^{4} − 4Y^{2}Y^{2} + 4Y^{8}
+ 70u^{4}u^{2} − 7u^{4}
+ 14u^{2}u^{2} + u^{2}) dx ^{m}mY^{k}I