Birkhoff–Poritsky Conjecture for Contact Billiards

Establish whether every hypersurface whose outer contact billiard correspondence is integrable must be a quadratic table.

Background

Quadratic tables are the principal completely integrable examples studied in the paper. The proposed Birkhoff–Poritsky analogue asks whether integrability characterizes quadratic tables among all hypersurfaces.

References

(Birkhoff-Poritsky ) Given a hypersurface $\Sigma$ such that the outer contact billiard correspondence is integrable, can we ensure $\Sigma$ is a quadratic table?

— Outer Contact Billiards  (2608.19393 - Caliz et al., 19 Aug 2026) in Section 6, “Open questions,” item 10(b), “Classical billiard conjectures”