Integrability by quadratures of the problem of rolling motion of a heavy homogeneous ball on a surface of revolution of the second order

In this paper, we consider the problem of rolling of a heavy homogeneous ball on a perfectly rough surface of revolution. Usually, when considering this problem, it is convenient to specify explicitly the surface along which the center of the ball moves during rolling, instead of the surface along which the ball rolls. The surface on which the center of the ball moves is equidistant to the surface on which the ball is rolling. It is well known, that the considered problem is reduced to the integration the second order linear homogeneous differential equation. In this paper we assume, that the surface along which the center of the ball moves is a non - degenerate surface of revolution of the second order. Using the Kovacic algorithm we prove that the general solution of the corresponding linear differential equation can be found explicitly. This means, that in this case the problem of rolling of a ball on a surface of revolution can be integrated by quadratures.