Perfect-fluid space-times with type-D Weyl tensor

We consider solutions of the Einstein field equations for which the Weyl tensor is of Petrov type D, and whose source is a perfect fluid with equation of state p=p(w), where p and w are the energy density and pressure of the fluid, respectively. We also impose two additional restrictions which are satisfied by most of the known solutions, namely, that the fluid 4-velocity u lies in the 2-space spanned by the two repeated principal null directions of the Weyl tensor, and that the Weyl tensor has zero magnetic part relative to u. Our main result is that for this class of solutions, the equation of state satisfies either dp/dw=0 or dp/dw= 1, or else the solution admits three or more Killing vector fields.