  Our aim is to study invariant hypersurfaces immersed in the Euclidean space
$\mathbb{R}^{n+1}$, whose mean curvature is given as a linear function in the
unit sphere $\mathbb{S}^n$ depending on its Gauss map. These hypersurfaces are
closely related with the theory of manifolds with density, since their weighted
mean curvature in the sense of Gromov is constant. In this paper we obtain
explicit parametrizations of constant curvature hypersurfaces, and also give a
classification of rotationally invariant hypersurfaces.
