Geometry and Real-Analytic Integrability

This note constructs a compact, real-analytic, riemannian 4-manifold ({\Sigma}, g) with the properties that: (1) its geodesic flow is completely integrable with smooth but not real-analytic integrals; (2) {\Sigma} is diffeomorphic to $T^2\times S^2$ ; and (3) the limit set of the geodesic flow on the universal cover is dense. This shows there are obstructions to realanalytic integrability beyond the topology of the configuration space.