Birkhoff–Poritsky conjecture (full generality)

Prove that the only integrable Birkhoff billiards in the plane are ellipses, thereby establishing the Birkhoff–Poritsky conjecture in full generality.

Background

The classical Birkhoff billiard is a planar billiard inside a strictly convex smooth closed curve. Birkhoff and Poritsky conjectured that integrability forces the boundary to be an ellipse. While many partial and local results are known, a complete proof for all smooth convex tables remains elusive.

The survey reviews several approaches (algebraic, variational/total integrability, KAM/local) and presents rigidity theorems under additional assumptions (e.g., central symmetry or total integrability on specific regions), but none settle the conjecture in full generality.