Research of the structure of the Liouville foliation of an integrable elliptical billiard with polynomial potential

In this paper we consider a planar billiard bounded by an ellipse in the potential force field. An explicit formula of the polynomial potential preserving integrability of such a billiard was found. The structure of the Liouville foliation at all non singular energy levels was studied using the method of separation of variables. Namely, an algorithm that constructs the bifurcation diagram and the Fomenko-Zieschang invariants from the values of the parameters of the potential was proposed. In addition, the topology of the isoenergetic manifold was studied and the cases of rigid body dynamics, which are Liouville equivalent to our billiard, were established.