The Reduction Method in the Theory of Lie-Algebraically Integrable Oscillatory Hamiltonian Systems

Abstract

We study the problem of the complete integrability of nonlinear oscillatory dynamical systems connected, in particular, both with the Cartan decomposition of a Lie algebra $G = K \oplus P{\text{, where }}K$ is the Lie algebra of a fixed subgroup $K \subset {\text{G}}$ with respect to an involution σ : G → G on the Lie group G, and with a Poisson action of special type on a symplectic matrix manifold.