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156 10 Generation techniquesmay be of the perfect fluid type, in which case2Φ ,a;b − 2Φ ,a Φ ,b + g ab Φ ,n Φ ,n = κ 0 (µ + p)u a u b + 1 3 κ 0pg ab (10.118)holds, cp. §3.7 (the covariant derivatives are understood with respect tog ab ). The integrability conditions for this system of differential equationsfor Φ have been studied by Kozameh et al. (1985) and in a series ofpapers by Van den Bergh. Obviously, all conformally flat perfect fluids(see Theorem 37.17) are contained here. The general solution of (10.117)is not known, but (for µ+p ̸= 0) the following results have been obtained.(1) If the fluid and vacuum solutions share a G r , r ≥ 2, or if the fluid hasan equation of state p = p(µ) and the conformal factor obeys Φ ,a u a =0,the solutions are shearfree or irrotational (Van den Bergh 1988b).(2) Shearfree solutions are conformally flat and thus covered by Theorem37.17, or of Petrov type D and locally rotationally symmetric andthus known (Van den Bergh 1986b, 1986e, Ellis 1967, Stewart and Ellis1968).(3) Irrotational fluids are conformally flat (if Φ ,a is parallel to u a ), orshearfree and non-expanding with a G 3 on V 2 or conformally related tothe Bianchi type VI 0 vacuum solution (13.58) (if Φ ,a is orthogonal to u a ),or expansionfree and admit a G 3 on V 2 (in the general case) (Van denBergh 1986d).(4) The vacuum solutions cannot be of Petrov type N, and Φ ,n cannotbe proportional to a Killing vector (Van den Bergh 1986c, 1987).The only null Einstein–Maxwell fields obtainable in a similar mannerare pp-waves (Van den Bergh 1986a). The conformally Ricci-flat pureradiation solutions are either pp-waves or contained in a Petrov type Nsolution in Kundt’s class, (31.38) with W o = 0 and (31.39) replaced by( )[](ζ + ¯ζ)H0H oζ + ¯ζ =(ζ +ia)+c.c. (10.119),ζ ¯ζ(ζ +ia)(¯ζ − ia)(Wils 1989a).The Einstein spaces conformal to pp-waves have been given by Siklos(1985) within a more general class including a pure radiation field; byTheorem 3.1, the pp-waves are necessarily non-vacuum.All these results are somewhat disappointing in that conformal transformationdid not really lead to new solutions. The same is true for attemptsto use five-dimensional backgrounds for a generation method (which leadsto equations very similar to (10.117) for the scalar field connected withg 55 ).,ζ