The classical Birkhoff conjecture states that the only integrable convex planar domains are circles and ellipses. In a joint work with A. Avila and J. De Simoi we show that this conjecture is true for perturbations of ellipses of small eccentricity. It turns out that the method of proof gives an insight into deformational spectral rigidity of planar axis symmetric domains and a partial answer to a question of P. Sarnak. The latter is a joint work with J. De Simoi and Q. Wei.