Empty singularities in higher-dimensional Gravity

We study the exact solution of Einstein's field equations consisting of a ($n+2$)-dimensional static and hyperplane symmetric thick slice of matter, with constant and positive energy density $\rho$ and thickness $d$, surrounded by two different vacua. We explicitly write down the pressure and the external gravitational fields in terms of $\rho$ and $d$, the pressure is positive and bounded, presenting a maximum at an asymmetrical position. And if $\sqrt{\rho}\,d$ is small enough, the dominant energy condition is satisfied all over the spacetime. We find that this solution presents many interesting features. In particular, it has an empty singular boundary in one of the vacua.