Topological analysis of an elliptic billiard in a fourth-order potential field

A planar billiard is considered in an elliptic domain in the case a polynomial potential of fourth degree acts to a material point. This dynamical system always has the first integral called the total energy which is also a Hamiltonian of this system. Assuming some additional conditions on the potential to guaranty the existence of another first integral which is independent on the Hamiltonian, the system turns out to be a Liouville integrable. The paper presents topological analysis of the corresponding Liouville foliation of this system. Namely, bifurcation diagrams are constructed and Fomenko–Zieschang invariants are calculated.