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.