Elliptical billiard table with Newtonian potential (Q1893654)

The problem of determining the motion of a particle moving on an elliptical billiard affected by an elastic force directed towards the center of the ellipse was considered in [(*) \textit{V. V. Kozlov}, J. Appl. Math. Mech. 52, No. 6, 691-699 (1988); translation from Prikl. Mat. Mekh. 52, No. 6, 883-894 (1988; Zbl 0711.70021)]. It was proved in (*) that the surface of such a billiard is integrable. Using this result, we will prove here that an elliptical billiard in which a particle moves under the effect of Newtonian force directed towards the focus of the ellipse is integrable.