The geometry of Poisson brackets and methods for integration, in the sense of Liouville, of systems on symmetric spaces

A survey is given of various methods of integrating Hamiltonian systems on symmetric spaces and Lie algebras, and an analysis is also presented of applications of the technique developed to some mechanical problems. Some questions of nonintegrability of Hamiltonian systems on symplectic manifolds are considered, and, in particular, the Morse theory of completely integrable Hamiltonian systems is expounded.