Non-smooth examples like the Bunimovich stadium are known. Smooth, strictly convex billiards are not ergodic. Most smooth Birkhoff billiards should have positive entropy. Problem probably first posed by Y. Sinai. Orbits computed with Mathematica.
Problem depends on the definition of integrability.
A real-analytic table with integral which is analytic in the angle must be an ellipse. Unclear whether it is a question of Birkhoff. Shows an orbit of a billiard in a table of equal width. The orbit is located on an invariant curve. Integrability would mean that every orbit behaves in an almost periodic way.