Topology of Liouville foliation for integrable billiard with gravitational potential bounded by parabolas

We study billiards in a gravitational field on a flat table bounded by arcs of confocal parabolas. Depending on the shape of the table, bifurcation diagrams are constructed and bifurcations of Liouville tori corresponding to their arcs are described. For non-singular energy zones, the Fomenko–Zieschang invariants are calculated.