Full rigidity conjecture for two-centre billiards

Establish that a bounded strictly convex planar domain containing the two Newtonian centres yields a two-centre billiard integrable at every energy level if and only if its boundary is an ellipse with foci at the two centres.

Background

The two-centre problem is Liouville-integrable in the interior and separates in elliptic–hyperbolic coordinates associated with the two fixed centres. When the reflecting boundary is a confocal ellipse, the separation integral is preserved by elastic reflection, producing an integrable two-centre billiard.

The conjecture seeks a geometric classification of all domains whose two-centre billiards are integrable at every energy. The paper proves only a real-analytic non-integrability result for non-confocal domains at each non-negative energy, under specified regularity assumptions; it does not resolve the conjecture for all notions of integrability or for the full stated class.

References

Let $\Omega\subset\mathbb{R}{2}$ be a bounded strictly convex domain containing the two centres $c_{1},c_{2}$. Then the two-centre billiard in $\Omega$ is integrable at every energy level if and only if $\partial\Omega$ is an ellipse with foci $c_{1}$ and $c_{2}$.

— Analytic rigidity and symbolic dynamics for two-centre billiards  (2609.02310 - Baranzini et al., 2 Sep 2026) in Conjecture, equation/label \ref{conj:main}, Introduction