  Using an orthonormal Lorentz frame approach to axistationary perfect fluid
spacetimes, we have formulated the necessary and sufficient equations as a
first order system, and investigated the integrability conditions of this set
of equations. The integrability conditions are helpful tools when it comes to
check the consequences and/or compatibility of certain simplifying assumptions,
e.g. Petrov types. Furthermore, using this method, a relation between the fluid
shear and vorticity is found for barotropic fluids. We collect some results
concerning Petrov types, and it is found that an incompressible axistationary
perfect fluid must be of Petrov type I.
