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.