Completely integrable torus actions on symplectic cones

We study completely integrable torus actions on symplectic cones (equivalently, completely integrable torus actions on contact manifolds). We show that if the cone in question is the punctured cotangent bundle of a torus then the action has to be free. From this it follows easily, using hard results of Marie and of Burago and Ivanov, that a metric on a torus whose geodesic flow admits global action-angle coordinates is necessarily flat thereby proving a conjecture of Toth and Zelditch.