Geometry and real-analytic integrability

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