  Spherical systems with polytropic equations of state are of great interest in
astrophysics. They are widely used to describe neutron stars, red giants, white
dwarfs, brown dwarfs, main sequence stars, galactic halos, and globular
clusters of diverse sizes. In this paper we construct analytically a family of
self-gravitating spherical models in the post-Newtonian approximation of
general relativity. These models present interesting cusps in their density
profiles which are appropriate for the modeling of galaxies and dark matter
halos. The systems described here are anisotropic in the sense that their
equiprobability surfaces in velocity space are nonspherical, leading to an
overabundance of radial or circular orbits, depending on the parameters of the
model under consideration. Among the family of models, we find the
post-Newtonian generalization of the Plummer and Hernquist models. A close
inspection of their equation of state reveals that these solutions interpolate
smoothly between a polytropic sphere in the asymptotic region and an inner core
that resembles an isothermal sphere. Finally, we study the thermodynamics of
these models and argue for their stability.
