Zero entropy, non-integrable geodesic flows and a non-commutative rotation vector

Let g be a 2-step nilpotent Lie algebra; we say g is non-integrable if, for a generic pair of points p, p' is an element of g*, the isotropy algebras do not commute: [g(p), g(p)'] not equal 0. Theorem: If G is a simply-connected 2-step nilpotent Lie group, g = Lie(G) is non-integrable, D < G is a cocompact subgroup, and g is a left-invariant Riemannian metric, then the geodesic flow of g on T*(D\G) is neither Liouville nor non-commutatively integrable with C-0 first integrals. The proof uses a generalization of the rotation vector pioneered by Benardete and Mitchell.