An integrable deformation of an ellipse of small eccentricity is an   ellipse
  The classical Birkhoff conjecture says that the only integrable convex domains are circles and ellipses. In the paper we show that this a version of this conjecture is true for small perturbations of ellipses of small eccentricity. 
Artur Avila, Jacopo De Simoi, Vadim Kaloshin
math.DS