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We use analytic perturbation theory to present a new approximate metric for arigidly rotating perfect fluid source with equation of state (EOS)$\epsilon+(1-n)p=\epsilon_0$. This EOS includes the interesting cases ofstrange matter, constant density and the fluid of the Wahlquist metric. It isfully matched to its approximate asymptotically flat exterior usingLichnerowicz junction conditions and it is shown to be a totally generalmatching using Darmois-Israel conditions and properties of the harmoniccoordinates. Then we analyse the Petrov type of the interior metric and showfirst that, in accordance with previous results, in the case corresponding toWahlquist'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 thestatic case) O and also that the non-static constant density case can only beof type I. Finally, we check that it can not be a source of Kerr's metric.

An approximate global solution of Einstein's equation for a rotating compact source with linear equation of state