Classification of Singularities of the Liouville Foliation of an Integrable Elliptical Billiard with a Potential of Fourth Degree

The paper is devoted to the study of a billiard bounded by an ellipse and equipped with a fourth degree potential as an integrable Hamiltonian system with two degrees of freedom. In previous works, the author described the structure of the Liouville foliation of such a system on nonsingular levels of the Hamiltonian in terms of Fomenko–Zieschang invariants: marked molecules and 3-atoms. Moreover, the dependence of the structure of the bifurcation diagram on the parameters of the potential has been established. The present work continues this study. Thus, the structure of the Liouville foliation in a neighborhood of critical layers containing a nondegenerate singular point of rank 0 or a degenerate orbit has been described. A classification of the obtained semilocal singularities was given. Finally, connections of our system with well-known cases of rigid body dynamics containing equivalent singularities is established. DOI 10.1134/S1061920823040155