  We consider billiard ball motion in a convex domain on a constant curvature
surface influenced by the constant magnetic field. We examine the existence of
integral of motion which is polynomial in velocities. We prove that if such an
integral exists then the boundary curve of the domain determines an algebraic
curve in $\mathbf{C}^3$ which must be nonsingular. Using this fact we deduce
that for any domain different from round disc for all but finitely many values
of the magnitude of the magnetic field billiard motion does not have Polynomial
in velocities integral of motion.
