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 $\mathbf{T}^2 \times \mathbf{S}^2$; 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.