  We consider nonholonomic geodesic flows of left-invariant metrics and
left-invariant nonintegrable distributions on compact connected Lie groups. The
equations of geodesic flows are reduced to the Euler-Poincare-Suslov equations
on the corresponding Lie algebras. The Poisson and symplectic structures give
raise to various algebraic constructions of the integrable Hamiltonian systems.
On the other hand, nonholonomic systems are not Hamiltonian and the integration
methods for nonholonomic systems are much less developed. In this paper, using
chains of subalgebras, we give constructions that lead to a large set of first
integrals and to integrable cases of the Euler-Poincare-Suslov equations.
Further, we give examples of nonholonomic geodesic flows that can be seen as a
restrictions of integrable sub-Riemannian geodesic flows.
