  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 S. 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.
