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22.2 Stationary cylindrically-symmetric fields 347and the two equivalent equationsα ′′ − (2a +4β ′ )α ′ +2β ′2 =0,(22.23b)α ′′ − 2aα ′ − 2α ′2 +4χ ′2 =0.(22.23c)Both can be solved by quadratures: one can prescribe β(ρ) and solve alinear first-order differential equation (22.23b) for α ′ , or prescribe α(ρ)and solve (22.23b) for β ′ or (22.23c) for χ ′ . Plane symmetry correspondsto a =0.Philbin (1996) chose coordinates with λ = χ and arrived atβ ′2 +2α ′ − a 0 α ′ e −α +(α ′′ + α ′2 )/2 =0, α = β + δ, (22.24)which again can be solved by a quadrature if α(ρ) is prescribed.Kramer (1988a) proposed taking the metric (22.4a) and using x ≡ 1 2 ln fas a new independent variable. IntroducingF (x) = µ +3p , y = dk2p dx = ˙k,the field equations readz = 1 WdWdx = ẆW , (22.25)ẏ =(1− yz)(Fy− 2), ż =(1− yz)(Fz − 2). (22.26)Using one or other of the above equations, several exact solutionshave been constructed, e.g. for (22.21) and τ ′′ = −e −2τ (Evans 1977),for (22.24) and e α+δ = ρ + c 1 ρ 3 + c 2 ρ 5 (Philbin 1996), for (22.26) andy = (az − b)/(dz − c) (Haggag and Desokey 1996), for (22.26) andz = (dy 2 + cy + b)/(y − a) (Haggag 1999). By making other ad hocassumptions, solutions have also been found by Davidson (1989b, 1990a,1990b), and Narain (1988).For an equation of state p =(γ − 1)µ, special cases have been givenby Evans (1977) (5p = µ) and by Teixeira et al. (1977a) (3p = µ). Thegeneral solution was found by Bronnikov (1979) asds 2 =e 2[β+χ+δ] dρ 2 +e 2β dϕ 2 +e 2χ dz 2 − e 2δ dt 2 ,(22.27a)β = γ − 23γ − 2α(ρ)+bρ, χ = β + aρ, δ = α(ρ) − bρ,4(γ − 1) 4(γ − 1)(some integration constants have been set zero by choice of coordinates),where α(ρ) satisfies the differential equationα ′′ + Aα ′2 − Bα ′ +2b 2 =0, A = 7γ2 +20γ − 128(γ − 1) 2 , B =2a + b 3γ − 2γ − 1 ,(22.27b)