An optical Hamiltonian and obstructions to integrability

If the geodesic flow of a compact Finslerian 3-manifold is completely integrable, and its singular set is a tame polyhedron, then the manifold's pi(1) is almost solvable ( Butler 2005 Topology 44 769-89). Is this true for non-commutatively integrable geodesic flows? This paper constructs non-commutatively integrable optical Hamiltonians on the unit-sphere bundle of homogeneous spaces of PSL2R that have a real-analytic singular set. These flows are not tangential to a Lagrangian foliation with a tame singular set.