  We examine various well known exact solutions available in the literature to
investigate the recent criterion obtained in ref. [20] which should be
fulfilled by any static and spherically symmetric solution in the state of
hydrostatic equilibrium. It is seen that this criterion is fulfilled only by
(i) the regular solutions having a vanishing surface density together with the
pressure, and (ii) the singular solutions corresponding to a non-vanishing
density at the surface of the configuration . On the other hand, the regular
solutions corresponding to a non-vanishing surface density do not fulfill this
criterion. Based upon this investigation, we point out that the exterior
Schwarzschild solution itself provides necessary conditions for the types of
the density distributions to be considered inside the mass, in order to obtain
exact solutions or equations of state compatible with the structure of general
relativity. The regular solutions with finite centre and non-zero surface
densities which do not fulfill the criterion [20], in fact, can not meet the
requirement of the `actual mass' set up by exterior Schwarzschild solution. The
only regular solution which could be possible in this regard is represented by
uniform (homogeneous) density distribution.
  The criterion [20] provides a necessary and sufficient condition for any
static and spherical configuration (including core-envelope models) to be
compatible with the structure of general relativity. Thus, it may find
application to construct the appropriate core-envelope models of stellar
objects like neutron stars and may be used to test various equations of state
for dense nuclear matter and the models of relativistic stellar structures like
star clusters.
