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608 38 The interconnections between the classification schemesG 2 . These latter solutions are: (i) the Demianski solution (29.62), (ii) thesolutionm +iM = m 0 +iM 0 = const, P 2 =(x √ 2) 3 =(ζ + ζ) 3 ,L = 1 6 iM 0(2) −1/2 x −3 [ C 1 x √ 13/2 + C 2 x −√ 13/2 ] (38.4)(C 1 ,C 2 being real constants), which is a special member of (29.50) towhich it is related by a transformation (29.26) with f = ζ, F ,u =1, and(iii) the solutionP =1, iM − m =2A(1 + α)ζ α ,L= Aζ 2 ζ 1+α + Bζζ α/3 , (38.5)where Re α = −3, and A and B are complex constants. The metrics(38.4) and (38.5) respectively admit groups G 2 I and G 2 II and belong tothe classes (29.46)–(29.50) and (29.60)–(29.61). A vacuum solution of theclass (29.46)–(29.50), with a G 2 II, was found by Lun (1978): it readsm +iM =(m 0 +iM 0 )ζ 3/2 , P 2 =(ζ + ζ) 3 , s ≡ y/x,(L = x[A−3/2 s + √ ) √1+s 2 13/2 ( s − 2√ 1 √ )13 1+s 2(38.6)+B(s + √ ) √− 13/2 (1+s 2 s + 2√ 1 √ ) ]13 1+s 2 + m +iM3P 2(m 0 ,M 0 ,A,B real constants). The twisting type N vacuum solutionsadmit at most a (non-Abelian) G 2 (Stephaniand Herlt 1985).The classes (29.46)–(29.50) and (29.60)–(29.61) cover all algebraicallyspecial diverging vacuum solutions which admit a G 1 generated by anasymptotically timelike Killing vector field ξ = ∂ u (Held 1976a, 1976b,Zenk and Das 1978). The algebraically special vacuum solutions (ρ ̸= 0)with an orthogonally transitive G 2 I (cp. §8.6) are of Petrov type D (Weirand Kerr 1977).The groups of motion of the Robinson–Trautman vacuum solutions(Chapter 28) were systematically analysed by Collinson and French (1967)using a null tetrad formulation of the Killing equations. The results aregiven in Table 38.2. To the authors’ knowledge, the groups of motions ofthe algebraically special diverging non-empty spaces have not been systematicallyinvestigated and the same is true for the symmetries of the(non-diverging) solutions of Kundt’s classs (Chapter 31) except for thenull Killing vector case and the pp-waves, see §§24.4–24.5 and Tables 24.1–24.2).Tables 38.3–38.5 give the solutions listed in this book for which both thePetrov type and the symmetries are known. Special cases of some solutionsmay admit a higher-dimensional group or/and the Petrov type may be