Stationary and Axisymmetric Perfect Fluids with one Conformal Killing   Vector
  We study the stationary and axisymmetric non-convective differentially rotating perfect-fluid solutions of Einstein's field equations admitting one conformal symmetry. We analyse the two inequivalent Lie algebras not exhaustively considered in Mars and Senovilla, 1994, and show that the general solution for each Lie algebra depends on one arbitrary function of one of the coordinates while a set of three ordinary differential equations for four unknowns remains to be solved. The conformal Killing vector of these solutions is necessarily homothetic. We summarize in a table all the possible solutions for all the allowed Lie algebras and also add a corrigendum to an erroneous statement in the paper quated above concerning the differentially rotating character of one of the solutions presented 
Marc Mars, Jose M. M. Senovilla