  We use a phase space analysis to give some classification results for
rotational hypersurfaces in $\mathbb{R}^{n+1}$ whose mean curvature is given as
a prescribed function of its Gauss map. For the case where the prescribed
function is an even function in $\mathbb{S}^n$, we show that a Delaunay-type
classification holds for this class of hypersurfaces. We also exhibit examples
showing that the behavior of rotational hypersurfaces of prescribed
(non-constant) mean curvature is much richer than in the constant mean
curvature case.
