Stationary Motion of a Self-gravitating Toroidal Incompressible Liquid Layer

We consider an incompressible fluid contained in a toroidal stratum which is only subjected to Newtonian self-attraction. Under the assumption of infinitesimal thickness of the stratum we show the existence of stationary motions during which the stratum is approximately a round torus (with radii $r$, $R$ and $R \gg r$) that rotates around its axis and at the same time rolls on itself. Therefore each particle of the stratum describes an helix-like trajectory around the circumference of radius $R$ that connects the centers of the cross sections of the torus.