  A description of Lagrangian and Hamiltonian formalisms naturally arisen from
the invariance structure of given nonlinear dynamical systems on the
infinite--dimensional functional manifold is presented. The basic ideas used to
formulate the canonical symplectic structure are borrowed from the Cartan's
theory of differential systems on associated jet--manifolds. The symmetry
structure reduced on the invariant submanifolds of critical points of some
nonlocal Euler--Lagrange functional is described thoroughly for both
differential and differential discrete dynamical systems. The Hamiltonian
representation for a hierarchy of Lax type equations on a dual space to the Lie
algebra of integral-differential operators with matrix coefficients, extended
by evolutions for eigenfunctions and adjoint eigenfunctions of the
corresponding spectral problems, is obtained via some special Backlund
transformation. The connection of this hierarchy with integrable by Lax
spatially two-dimensional systems is studied.
