On the Birkhoff Conjecture for Kepler Billiards

The paper proves, via a robust symbolic-dynamics construction at high energy and a rigidity result about focal points, that Kepler billiards are not analytically integrable except (i) possibly at one center for non-elliptic tables and (ii) exactly at the two foci for ellipses, matching its Theorem 1.1. The model’s outline reaches the same headline conclusion but relies on an unsubstantiated analytic-limit argument (Montel + passage of first integrals to the free billiard) and on conflating analytic first integrals with the Poritsky/Liouville foliation property—precisely the distinction the paper highlights as non-equivalent. It also sketches Melnikov/splitting claims without proof. Therefore, the paper’s result stands; the model’s proof is not sound.