Birkhoff Conjecture for nearly centrally symmetric domains

In this paper we prove a perturbative version of a remarkable Bialy-Mironov (Ann.Math:389-413(196), 2022) result. They prove non perturbative Birkhoff conjecture for centrally-symmetric convex domains, namely, a centrally-symmetric convex domain with integrable billiard is ellipse. We combine techniques from Bialy-Mironov (Ann.Math:389-413(196), 2022) with a local result by Kaloshin-Sorrentino (Ann.Math:315-380(188), 2018) and show that a domain close enough to a centrally-symmetric one with integrable billiard is ellipse. To combine these results we derive a slight extension of Bialy-Mironov (Ann.Math:389-413(196), 2022) by proving that a notion of rational integrability is equivalent to the $C^0$-integrability condition used in their paper.