Some results on the integrability of Einstein's field equations for axistationary perfect fluids

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.