  We model a rotating star as a compressible fluid subject to gravitational
forces. In almost all the mathematical literature the entropy is considered to
be constant. Here we allow it to be variable. We consider a star that steadily
rotates differentially around a fixed axis, say the $z$-axis. We prove the
existence of a family of such stars with small angular velocity $\omega$ and
small entropy variation $s$ and with an equation of state $p=Ke^s\rho^\gamma$.
Our analysis reduces to a hyperbolic equation for the modified entropy coupled
to an elliptic equation for the modified density, together with a mass
constraint. Due to the variable entropy and the consequent loss of both
regularity and variational structure, all the methods in the previous
literature fail. We develop a new ad hoc perturbative strategy that allows us
to construct rotating stars that bifurcate from the non-rotating ones.
