INTEGRABILITY ANALYSIS OF A CONFORMAL EQUATION IN RELATIVITY

In 1987 C. C. Dyer, G. C. McVittie, and L. M. Oattes derived the (two) field equations for shear-free, spherically symmetric perfect fluid s pacetimes which admit a conformal symmetry. We use the techniques of t he Lie and Painleve analyses of differential equations to find solutio ns of these equations. The concept of a pseudo-partial Painleve proper ty is introduced for the first time which could assist in Ending solut ions to equations that do not possess the Painleve property. The pseud o-partial Painleve property throws tight on the distinction between th e classes of solutions found independently by P. Havas and M. Wyman. W e find a solution for all values of a particular parameter for the fir st field equation and link it to the solution of the second equation. We indicate why we believe that the first field equation cannot be sol ved in general. Both techniques produce similar results and demonstrat e the close relationship between the Lie and Painleve analyses. We als o show that both of the field equations of Dyer er al. may be reduced to the same Emden-Fowler equation of index two.