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31.8 Solutions includinga cosmological constant Λ 483(31.41), these solutions admit a group of motions G 4 on T 3 (cp. (13.48)).Unlike the vacuum case this class does not exhaust all type D solutions: itdoes not, for instance, contain an Einstein–Maxwell field given by KowalczyńskiandPlebański(1977) in the formds 2 =2x −2 [(dx/A) 2 + A 2 dy 2 +(dz/B) 2 − B 2 dt 2 ],A 2 = ax 2 + cx 3 − 2e 2 x 4 , B 2 = b − az 2 , a,b,c,e= const.(31.60)The Einstein–Maxwell type D solutions (double aligned) of Kundt’sclass have been determined by Plebański(1979). They are given byds 2 = x2 + l 2K(x) dx2 + K(x) [x 2 + l 2 dσ +]vdu − udv+2 (x2 + l 2 )1 − εuv (1 − εuv) 2 dudv,K(x) =2nx − (e 2 + g 2 ) − 2ε(x 2 − l 2 ), (31.61)Φ 11 =(e 2 + g 2 )/2(x 2 + l 2 ) 2 ,e,g,l,ε,n= const,and contain the solutions (24.21)–(24.22) – with Λ = 0 – with a group G 2on null orbits.As in the vacuum case, Theorem 31.1 can be used to generate newsolutions from known ones.31.8 Solutions including a cosmological constant ΛIf we want to include (add) an energy-momentum tensor −Λg ik , cp. (5.4),then we have to replace the field equations (31.16b) byR 12 =2κ 0 Φ 1 Φ 1 +Λ, R 34 =2κ 0 Φ 1 Φ 1 − Λ, (31.62)the rest of the field equations remaining unchanged. No general theoremsaying how to incorporate Λ into a vacuum or Einstein–Maxwell field isavailable, but solutions have been found in a number of subcases.Starting from type III and N vacuum solutions, one has ∆ ln P =Λ,i.e. P = 1+Λζζ/2 instead of P = 1, and H ,vv = − 1 2 P 2 W ,v W ,v − Λinstead of (31.29). The type N solutions falling into this class have beengiven by Ozsváth et al. (1985) (see also García D. and Plebański(1981)and Bičák and Podolsky (1999)), and the solutions with W ,v = 0 byLewandowski(1992).Metrics for which the multiple null eigenvector k is recurrent, k a;b =k a p b , have been studied by Leroy and McLenaghan (1973).Some special type II solutions with constant Φ 1 Φ 1 have been constructedby García D. and Alvarez C. (1984) and Khlebnikov (1986).