On the Integrability of Strictly Convex Billiard Tables with Boundaries Close to Ellipses with Small Eccentricities

In this paper, we introduce a new notion of integrability for billiard tables, namely, integrability away from the boundary. One key feature of our notion is that the integrable region could be disjoint from the boundary with a positive distance. We prove that if a strictly convex billiard table, whose boundary is a small perturbation of an ellipse with small eccentricity, is integrable in this sense, then its boundary must be itself an ellipse.