  Circularly rotating axisymmetric perfect fluid space-times are investigated
to second order in the small angular velocity. The conditions of various
special Petrov types are solved in a comoving tetrad formalism. A number of
theorems are stated on the possible Petrov types of various fluid models. It is
shown that Petrov type II solutions must reduce to the de Sitter spacetime in
the static limit. Two space-times with a physically satisfactory
energy-momentum tensor are investigated in detail. For the rotating
incompressible fluid, it is proven that the Petrov type cannot be D. The
equation of the rotation function $\omega $ can be solved for the Tolman type
IV fluid in terms of quadratures. It is also shown that the rotating version of
the Tolman IV space-time cannot be Petrov type D.
