Integrable Geodesic Flows on Cones over Riemannian Manifolds

In this paper we study the behavior of geodesics on cones over arbitrary $C^3$-smooth closed Riemannian manifolds. We show that the geodesic flow on such cones admits first integrals whose values uniquely determine almost all geodesics except for radial geodesics; thus, the geodesic flow is superintegrable. Moreover, we prove that the geodesic flow restricted to the open dense subset of the cotangent bundle corresponding to all non-radial trajectories is Liouville--Arnold integrable. This investigation is inspired by our recent results on Birkhoff billiards inside cones over convex manifolds where similar results hold true.