  The classical Birkhoff conjecture claims that the boundary of a strictly
convex integrable billiard table is necessarily an ellipse (or a circle as a
special case). In this article we prove a complete local version of this
conjecture: a small integrable perturbation of an ellipse must be an ellipse.
This extends and completes the result in [3], where nearly circular domains
were considered. One of the crucial ideas in the proof is to extend
action-angle coordinates for elliptic billiards into complex domains (with
respect to the angle), and to thoroughly analyze the nature of their complex
singularities. As an application, we are able to prove some spectral rigidity
results for elliptic domains.
