Slowly rotating fluid balls with linear equation of state
  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. 
Gyula Fodor