General integrability classification for two-centre billiards

Characterize whether integrability of a two-centre billiard at every energy level, under any standard notion of billiard integrability, forces the boundary to be a confocal ellipse with foci at the two centres.

Background

The authors deliberately leave the meaning of integrability broad, encompassing such possibilities as a non-constant first integral with prescribed regularity and a foliation of the billiard-map domain by invariant curves.

This formulation makes the unresolved issue stronger than the analytic first-integral theorem proved in the paper. The paper establishes the conjectured rigidity only for real-analytic first integrals in the non-negative-energy regime, not for every regularity class or every standard definition of integrability.