
As is well known symmetry analysis of nonlinear dynamical
systems on a smooth manifold gives rise in many cases to exhibiting its
many hidden but interesting properties, in particular such as being
integrable by quadratures due to the LiouvilleArnold theorem. In case when
the manifold can be represented as the cotangent space to some subgroup
of a Lie group naturally acting on it, the study of the corresponding flow
can be recast via the reduction method into the Hamiltonian framework due
to the existence on the canonical Poisson structure. Furthermore, if the
symmetry group naturally generalizes to the loop group over, then the
corresponding momentum mapping provides us with a Lax type representation
and related with it a complete set of commuting invariants. Such a scheme
appeared to be very useful when proving the Liouville integrability of many
finitedimensional systems such as Kowalevskaya's top, Neumann type systems
and other. In the report we study complete integrability of nonlinear
oscillatory dynamical systems connected in particular both with the Cartan
decomposition of a Lie algebra and with a Poisson action of special
type on a symplectic matrix manifold.
