On integrability of certain rank 2 sub-Riemannian structures

We discuss the integrability of rank 2 sub-Riemannian structures on low-dimensional manifolds, and then prove that some structures of that type in dimension 6, 7 and 8 have a lot of symmetry but no integrals polynomial in momenta of low degrees, except for those coming from the Killing fields and the Hamiltonian, thus indicating non-integrability of the corresponding geodesic flows.