<p>In this paper we address the problem of a body of revolution moving on a plane within the framework of the rubber rolling model. By “rubber” rolling we mean the rolling of the body without slipping at the point of contact and without twisting relative to the vertical. We write the equations of motion in local coordinates and reduce them to Hamiltonian form. We prove the isomorphism of the reduced systems for bodies of revolution whose surfaces are equidistant to each other. As an example, we consider the problem of a torus of revolution rolling on a plane, for which we carry out a complete bifurcation analysis of partial solutions. Also, we analyze the dynamics of the torus in absolute space and perform a classification of trajectories of motion. In addition, we touch upon the question of whether a retrograde turn is possible as the torus rolls on the plane.</p>