Status of self - bound structures in hydrostatic equilibrium The actual meaning of exterior Schwarzschild solution

We show that regular structures governed by an equation of state (EOS) or density variation such that the density at the surface does not vanish together with pressure (also known as self-bound regular structures) can not exist in the state of hydrostatic equilibrium. The only configuration which could exist in this regard is governed by homogeneous density distribution. Other structures which can fulfill the requirement set up by exterior Schwarzschild solution (and, therefore, can exist in hydrostatic equilibrium) would always be governed by a singular solution (that is, the pressure and density both become infinity at the centre), and in this sense can not be considered regular.