Topology of integrable billiard in an ellipse on the Minkowski plane with the Hooke potential

The integrability of billiards bounded by arcs of confocal quadrics in the Minkowski plane in a field with the Hooke potential is obtained. The case of this type of a billiard in an ellipse is studied in detail. The topology of Liouville foliations arising in this problem is also studied and Fomenko invariants are also constructed.