  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.
