  Cataldo has found all rigidly rotating self-gravitating perfect fluid
solutions in 2+1 dimensions with a negative cosmological constant $\Lambda$,
for a density that is specified a priori as a function of a certain radial
coordinate. We rewrite these solutions in standard polar-radial coordinates,
for an arbitrary barotropic equation of state $p(\rho)$. For any given equation
of state, we find the two-parameter family of solutions with a regular centre
and finite total mass $M$ and angular momentum $J$ (rigidly rotating stars).
For analytic equations of state, the solution is analytic except at the
surface, but including at the centre. Defining the dimensionless spin $\tilde
J:=\sqrt{-\Lambda}\,J$, there is precisely one solution for each $(\tilde J,M)$
in the region $|\tilde J|-1<M<|\tilde J|$, which consists of parts of the point
particle region $M<-|\tilde J|$ and overspinning regions $|\tilde J|>|M|$. In
an adjacent compact part of the black hole region $|\tilde J|<M$ (whose extent
depends on the equation of state), there are precisely two solutions for each
$(\tilde J,M)$. Hence exterior solutions exist in all three classes of BTZ
solution (black hole, point particle and overspinning), but not all possible
values of $(\tilde J,M)$ can be realised as stars. Regardless of the values of
$\tilde J$ and $M$, the causal structure of all stars for all equations of
state is that of anti-de Sitter space, without horizons or closed timelike
curves.
