Examples of integrable sub-Riemannian geodesic flows

Motivated by a paper of Bolsinov and Taimanov DG/9911193 we consider non-holonomic situation and exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy. Moreover the Riemannian examples are obtained as "holonomization" of sub-Riemannian ones. A feature of non-holonomic situation is non-compactness of the phase space. We also exhibit a Liouvulle-integrable Hamiltonian system with topological entropy of all integrals positive.