  We use analytic perturbation theory to present a new approximate metric for a
rigidly rotating perfect fluid source with equation of state (EOS)
$\epsilon+(1-n)p=\epsilon_0$. This EOS includes the interesting cases of
strange matter, constant density and the fluid of the Wahlquist metric. It is
fully matched to its approximate asymptotically flat exterior using
Lichnerowicz junction conditions and it is shown to be a totally general
matching using Darmois-Israel conditions and properties of the harmonic
coordinates. Then we analyse the Petrov type of the interior metric and show
first that, in accordance with previous results, in the case corresponding to
Wahlquist's metric it can not be matched to the asymptotically flat exterior.
Next, that this kind of interior can only be of Petrov types I, D or (in the
static case) O and also that the non-static constant density case can only be
of type I. Finally, we check that it can not be a source of Kerr's metric.
