Invariant fibrations of geodesic flows

Let (Σ,g) be a compact C2 finslerian 3-manifold. If the geodesic flow of g is completely integrable, and the singular set is a tamely-embedded polyhedron, then π1(Σ) is almost polycyclic. On the other hand, if Σ is a compact, irreducible 3-manifold and π1(Σ) is infinite polycyclic while π2(Σ) is trivial, then Σ admits an analytic riemannian metric whose geodesic flow is completely integrable and singular set is a real-analytic variety. Additional results in higher dimensions are proven.