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348 22 Groups G 2 I on spacelike orbits: cylindrical symmetrywhich can easily be solved in terms of elementary functions. The solutionswith a =0,β= χ and α linear in ρ admit an additional homothetic vector(Debever and Kamran 1982). In the notation of Kramer (1988a), see alsoHaggag (1989), the solutions regular at the axis ρ = 0 are given byds 2 = F m dρ 2 + F n (F dz 2 + ρ 2 dϕ 2 ) − F s dt 2 ,κ 0 p =16(γ − 1)2 β(2 − γ)(7γ − 6) F −m−1 , F =1− βρ 2 ,(22.28)n = − 3γ − 27γ − 6 ,2nγm = −1 −γ − 2 , s = −4nγ − 1γ − 2 .If the four-velocity u = w −1 (ρ)∂ t is hypersurface-orthogonal, but thetwo spacelike Killing vectors are not, one can start from a metric of theform (22.4b), i.e. fromds 2 =(dy + ρdϕ)2h 2 (ρ)+ E(ρ)h 2 (ρ) dϕ2 + e−b(ρ) dρ 2h 2 (ρ)E(ρ) − w2 (ρ)dt 2 . (22.29)The field equations then give (Stephani 1998)E(ρ) =−ρ 2 + aρ +c , w 2 (ρ) =e −b(ρ) h 2 (ρ), (22.30)where h(ρ) and b(ρ) have to obeyh 2 b ′′ +4hh ′ b ′ − 4h ′2 =0. (22.31)The solutions of this differential equation can be given in either of thetwo forms∫∫∫ √bb = b ′ dρ, b ′ =4h −4 h 2 h ′2 dρ, or ln h = 1 2 b ± 1 2′′ + b ′2 dρ.(22.32)So one can prescribe h(ρ) orb(ρ), and determine the second function from(22.32). Pressure p and energy density µ are given byκ 0 p = − 1 4 eb h 2 [ 1+(b ′ E) ′] [ , κ 0 µ =e b h 2 2(Eh ′ /h) ′ − 1 (4 Eb′ ) ]′ +34.(22.33)The special case h = ρ 1/3 ,b=lnρ 4/3 admits a Killing tensor (Papacostas1988). For the metrics (22.29)–(22.30) there is a linear relation betweenthe metric coefficients of the y–ϕ-part, cp. §17.1.2; for c + a 2 /4 > 0 thereare two hypersurface-orthogonal Killing vectors and the solutions can betransformed into the form (22.20).The metrics (36.22) and (36.23) also admit an Abelian G 3 on T 3 .