Geometry and integrability of Euler–Poincaré–Suslov equations

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–Poincaré–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–Poincaré–Suslov equations. Further, we give examples of nonholonomic geodesic flows that can be seen as a restrictions of integrable sub-Riemannian geodesic flows.