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30.6 Remarks concerningsolutions of the different Petrov types 461generalizes (29.39). Any solution P, β of this equation together withL =0=M, Φ 0 1 = b 0β + ψ 0 , Φ 0 2 = P β ′ ,m = au − 2κ 0 b 0 ββ − 4κ 0 Re (ψ 0 β)+m 0 ,(30.33)gives a non-twisting Einstein–Maxwell field, in which one can hope toinsert twist (L ̸= 0,M ̸= 0) and an independent additional charge Φ 0 1 bysolving the last surviving equation (30.19c), with L and m +iM givenby (30.30)–(30.31). The additional charge ψ 0 can always be switched onor off.In §28.2, we have listed the known classes of solution of the backgroundequation (30.32). In each case twisting Einstein–Maxwell fields can befound, which generalize known twisting vacuum solutions, e.g. (29.46)–(29.50). In general, they are of Petrov type II or III. We refer the readerto the original paper (Herlt and Stephani 1984), but want to mention onespecial case given byP =1, Φ 0 1 = ψ(ζ), Φ0 2 = b, bL = ζψ′ − ψ +iˆB(ζ,ζ) ,ζ ,m +iM = −κ 0 b 2 [u +iˆB] − 2κ 0 b Re (ζψ),(30.34)where ˆB(ζ,ζ) is a real function satisfyingˆB ,ζζζζ= −b 2 ˆB (30.35)(whose general solution can be constructed, cp. the discussion of (30.66)).The existence of a constant electromagnetic pure radiation field impliesin m a term linear in u, i.e. a constant rate of loss of the mass m of thesystem.The field equations can also be simplified under the assumptions P ,u =(m +iM) ,u =Φ 0 1,u =Φ0 2,u =0,Llinear in u (Nurowskiand Tafel 1992),but no solutions have been found for L ,u ̸= 0.30.6 Remarks concerning solutions of the different PetrovtypesAll type D Einstein–Maxwell fields for which both null eigenvectors ofthe Maxwell field are multiple eigenvectors of the Weyl tensor are known.They depend on at most six arbitrary parameters; for a detailed discussion,see §21.1.2. The charged Kerr–NUT metrics covered by Theorem30.2 and the charged and twisting C-metric are included here. Besidesthese solutions, one more type D solution can be generated by means of