Integrable systems and toric contact forms on ℝℙ3

It is well-known that the geodesic flow on ellipsoids of revolution is integrable. In joint work with Ferreira and Vicente, we used this fact to obtain a symplectomorphism between the unit disk bundle of such an ellipsoid without fiber and a toric domain. In this talk, I will explain this result and how we can also obtain a symplectomorphism between the whole unit disk bundle and a toric filling of ℝℙ3, which can be concave or convex depending on the original ellipsoid. I will also explain how to generalize this idea to other situations.