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36.4 Solutions with other intrinsic symmetries 57936.4 Solutions with other intrinsic symmetriesMartinez and Sanz (1985) studied metrics of the form[ds 2 = B(t, r)dr 2 + C(t, r) dϑ 2 + M 2 (ϑ)dϕ 2] − A(t, r, ϑ, ϕ)dt 2 , (36.32)which on t = const have (at least) one Killing vector ∂ ϕ .Vacuum solutions of this form have either a G 3 on S 2 (the ϑ–ϕ-spacehas constant curvature) or are given by the special caseds 2 = t −2 dr 2 + a 2 t 2 (e 2x2 dx 2 + x 2 dy 2 ) − e 2x2 dt 2 , a = const (36.33)of (17.13) (admitting a G 2 ).Perfect fluid solutions with four-velocity orthogonal to t = const containthe conformally flat solutions (37.39) and (for A ϕ = 0) the two rigidlyrotating solutionsds 2 =dr 2 +dϑ 2 + F (ϑ) 2 ,ϑ dϕ2 − F 2 dt 2 ,F ,ϑϑ = b/F, κ 0 µ = κ 0 p = bF −2 , b = const(admitting a G 3 ) and(36.34)ds 2 =dr 2 /(λ + br 2 )+r 2 (dϑ 2 + F (ϑ) 2 ,ϑdϕ 2 ) − r 2 F 2 dt 2 ,(36.35)FF ,ϑϑ = c − λF 2 , κ 0 µ = c/r 2 F 2 − 3b = κ 0 p − 6b, λ, b, c = const(admitting a G 2 ), and the metricds 2 = t −2 dr 2 + t 2 (dϑ 2 + F (ϑ) 2 ,ϑdϕ 2 ) − (b ln t + c) −1 F 2 dt 2 ,F ,ϑϑ = aF −1 + bF −1 ln F, a, b, c = const, (36.36)κ 0 p =(tF ) −2 [a + b ln F − c − b ln t] =κ 0 µ + b(tF ) −2(admitting a G 2 ).In generalizing (36.32), Argüeso and Sanz (1985) considered the metricsds 2 = B(t, r)dr 2 + C(t, r)dϑ 2 + H(t, r)M 2 (ϑ)dϕ 2 − A(t, r, ϑ, ϕ)dt 2 ,(36.37)(H ̸= C), again with one symmetry (∂ ϕ )ont = const. If A ,ϕ ̸= 0,the solutions are either static degenerate vacuum B-metrics (with Λ) orthe conformally flat perfect fluid (37.39). If A ,ϕ =0, only the subcaseC = C(t), H = H(t) has been considered. It leads to solutions with aG 2 I and – for perfect fluids – to an equation of state µ = p, see §23.1.Space-times with a G 3 transitive on a S 3 , where the S 3 admits anintrinsic G 4 , have been studied by Szafron (1981), see also McManus(1995).