In this paper, the dynamics of a loose wheel is considered rolling on a horizontal plane, where the tread profile of the wheel is ellipsoidal. From the equations of motion, a two-parametric family of steady motions are obtained, corresponding to the rolling motion on a circular or straight line with a uniform rotational speed and tilt angle. It is shown that compared to the geometries analysed in the literature, the elliptical tread profile leads to a significant change in the qualitative behaviour of the wheel.