Rolling of a symmetric sphere on a horizontal plane without sliding or spinning

In this paper we perform a complete study of the dynamics of a symmetric sphere rolling on a horizontal plane without sliding or spinning. Integrals of motion that completely determine the behaviour of this systems in terms of elementary functions are explicitly written. Equilibrium points and closed orbits are systematically described. Our approach is geometric and we find that the system is equivalent to an ODE on the manifold S2 × S1.