  The Tolman-Oppenheimer-Volkoff (TOV) equation admits singular solutions in
addition to regular ones. Here, we prove the following theorem. For any
equation of state that (i) is obtained from an entropy function, (ii) has
positive pressure and (iii) satisfies the dominant energy condition, the TOV
equation can be integrated from a boundary inwards to the center. Hence,
thermodynamic consistency of the EoS precludes pathological solutions, in which
the integration terminates at finite radius (because of horizons, or
divergences / zeroes of energy density). At the center, the mass function
either vanishes (regular solutions) or it is negative (singular solutions). For
singular solutions, the metric at the center is locally isomorphic to
negative-mass Schwarzschild spacetime. This means that matter is stabilized
because the singularity is strongly repulsive. We show that singular solutions
are causally well behaved: they are bounded-acceleration complete, and they are
conformal to a globally hyperbolic spacetime with boundary. Finally, we show
how to modify unphysical equations of state in order to obtain non-pathological
solutions, and we undertake a preliminary investigation of dynamical stability
for singular solutions.
