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26.4 Algebraically general solutions 415The Newman–Tamburino solutions do not contain arbitrary functionsof time; the Robinson–Trautman solutions are not included. The metric(26.21) admits at most one Killing vector (ξ = ∂ u if b = 0); there is onlyone Killing vector (ξ = ∂ y ) for the metric (26.22), whereas (26.23) admitsan Abelian G 2 , the ignorable coordinates being y and u (Collinson andFrench 1967).Vacuum metrics where the null congruence with κ =0,ω =0,σ ̸=0, Θ ̸= 0is not a principal null congruence (Ψ 0 ̸= 0) have been consideredby Bilge (1989) under the assumption D(σ/ρ) =0. It turns out thata = σ/ρ = const (real), Ψ 0 = a(σ 2 − ρ 2 ). (26.24)For a ̸= 0, ±1, ±2, ±1/2, the Newman–Penrose equations could besolved. In coordinates x 1 = x, x 2 = y, x 3 = r, x 4 = u, the solutionsare of the formg 34 = −1, g nm =(ξ n ξ m + ξ n ξ m ), n,m =1, 2,g 33 = −2U, g 3i = −X i ,i=1, 2,(26.25a)withξ 1 = r −a+ , ξ 2 =ir −a− , X 1 = a + xψ ′ (u),X 2 = a − yψ ′ , U = −rψ ′ +(1+a 2 )r 1−c µ 0 ,a ± =(1± a)/(1 + a 2 ), c =2/(1 + a 2 ),⎧1 for ψ ⎪⎨′ =0,a − ⎫̸= 1/2,⎪⎬µ 0 = exp [−(c +1)ψ] for ψ ′ ̸= 0,a − ̸= 1/2,⎪⎩y for ψ ′ =0,a − =1/2 (special sol. only),⎪⎭(26.25b)(for µ 0 = 1 we have a Kasner-type metric with three Killing vectors andone homothetic vector).Among the solutions of the type (26.25) are the Kóta et al. (1982) solutions(for ψ ′ =0,a − =1/2 ) and type N solutions (for ψ ′ =0,a − = 0).A study of the case D(σ/σ) = 0 can be found in Bilge (1991). Algebraicallygeneral solutions with real ρ and σ and Ψ 0 = −3σ 2 /r admittwo spacelike commuting Killing vectors, see Bilge (1990).