Geometry and integrability of Euler-Poincare-Suslov equations

We consider non-holonomic geodesic flows of left-invariant metrics and left -invariant non-integrable distributions on compact connected Lie groups. Th e equations of geodesic flows are reduced to the Euler-Poincare-Suslov equa tions on the corresponding Lie algebras. The Poisson and symplectic structu res give rise to various algebraic constructions of the integrable Hamilton ian systems. On the other hand, non-holonomic systems are not Hamiltonian a nd the integration methods for non-holonomic systems are much less develope d. In this paper, using chains of subalgebras, we give constructions that l ead to a large set of first integrals and to integrable cases of the Euler- Poincare-Suslov equations. Furthermore, we give examples of non-holonomic g eodesic flows that can be seen as a restriction of integrable sub-Riemannia n geodesic flows.