Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane

We consider a convex curve gamma lying on the Sphere or Hyperbolic plane. We study the problem of existence of polynomial in velocities integrals for Birkhoff billiard inside the domain bounded by gamma. We extend the result by gamma. Bolotin (1992) and get new obstructions on polynomial integrability in terms of the dual curve Gamma. We follow a method which was introduced by S. Tabachnikov for Outer billiards in the plane and was applied later on in our recent paper to Birkhoff billiards with the help of a new the so called Angular billiard. (C) 2016 Elsevier B.V. All rights reserved.