Stationary perfect fluid solutions with differential rotation

In the framework of a class of metrics allowing for a timelike and a spacelike symmetries, usually referred to as stationary and axisymmetric gravitational fields, coupled to differentially rotating perfect fluids, the existing exact solutions to the corresponding Einstein equations are analyzed from the point of view of the fulfillment of the energy conditions and the existence of an axis of rotation. The main conclusion is that none of the reported exact solutions fulfills all these physical requirements, and at most, they can be thought of as stationary cyclic symmetric cosmological spacetimes.