  The Birkhoff conjecture says that the boundary of a strictly convex
integrable billiard table is necessarily an ellipse. In this article, we
consider a stronger notion of integrability, namely integrability close to the
boundary, and prove a local version of this conjecture: a small perturbation of
an ellipse of small eccentricity which preserves integrability near the
boundary, is itself an ellipse. This extends the result in [1], where
integrability was assumed on a larger set. In particular, it shows that (local)
integrability near the boundary implies global integrability. One of the
crucial ideas in the proof consists in analyzing Taylor expansion of the
corresponding action-angle coordinates with respect to the eccentricity
parameter, deriving and studying higher order conditions for the preservation
of integrable rational caustics.
