Are there any models with homogeneous energy density?

By applying a recent method --based on a tetrad formalism in General Relativity and the orthogonal splitting of the Riemann tensor-- to the simple spherical static case, we found that the only static solution with homogeneous energy density is the Schwarzschild solution and that there are no spherically symmetric dynamic solutions consistent with the homogeneous energy density assumption. Finally, a circular equivalence is shown among the most frequent conditions considered in the spherical symmetric case: homogeneous density, isotropy in pressures, conformally flatness and shear-free conditions. We demonstrate that, due to the regularity conditions at the center of the matter distribution, the imposition of two conditions necessarily leads to the static case.