  Slowly rotating perfect fluid balls with regular center and asymptotically
flat exterior are considered to second order in the rotation parameter. The
necessary condition for being Petrov type D is given for general perfect fluid
matter. As a special case, fluids with a linear equation of state are
considered. Using a power series expansion at the regular center, it is shown
that the Petrov D condition is inconsistent with the linear equation of state
assumption.
