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23.3 Solutions with a G 2 on S 2 371assumed that the G 2 I was an invariant subgroup in the three-parameterconformal group. Carot et al. (1996) gave some families only up to differentialequations or systems, and found the following explicit solutions.First, in comoving coordinates,ds 2 = S −2 [−dt 2 +dx 2 + (cosh x) 2m (e −2t dy 2 + sinh 2 x dz 2 )],(23.39a)S = b cosh ku(e −u/2 cosh x) m+1 , u ≡ t + ln cosh x, 4k 2 =1+m 2 ,(23.39b)κ 0 µ = 3Q2 − mS 2 cosh 2 x ,κ −Q(2m +3Q)0p =S 2 cosh 2 x,Q≡ (ln S) ,u +1; b, m, const.Mars and Wolf (1997) found the analogous solution with cosh x and sinh xexchanged, and one with sinh ku and cosh ku also exchanged.Secondly, using the comoving coordinates of Mars and Wolf (1997),ds 2 = S −2 (−dt 2 +dx 2 /x 2 + x b(b+1) cosh 1−b t dy 2 + x b(b−1) cosh b+1 t dz 2 ),S = x k + s 0 | sinh t| k , 2k =1+b 2 , b, s 0 const,(23.40)κ 0 µ = A + B, A ≡ 1 4 (b2 − 1)S 2 /(cosh t) 2 ,B≡ 3 4 (b2 +1) 2 (sinh t) b2−1 ,κ 0 p = A + C, 3(b 2 +1)C ≡ B[2(b 2 − 1)(x/ sinh t) k − b 2 − 5],together with the similar metric with cosh t and sinh t interchanged. Ifb = 1 this is conformally flat and if b = 0 it is Petrov type D. Ifs 0 → 0,this is a member of the class (23.26).The further solutions given by Carot et al. (1996) are: the subcasem = 0 of (23.13a), due to Allnutt (1980); a tilted fluid solution separablein non-comoving coordinates which, with constants a, b and c and in thenotation of (23.31), is given by 2T 1 = T 2 = a − (ln | sinh(2bct)|)/c, T 3 =0,e X 1= (cosh bx) (1−2c) , e X 2= sinh 2bx, e 2X 3= tanh bx, (23.41)(here the sinh in T 2 can be replaced by cosh or exp); a counterpart of(23.41) with t ↔ x; and a solution conformal to a decomposable spacetime(35.29) with E = 0, which therefore is of Petrov type D and has asix-parameter conformal group,ds 2 = a2F 2 [−dt 2 /(1 − t 2 )+dx 2 /(1 + x 2 )+(1− t 2 )dy 2 +(1+x 2 )dz 2] ,κ 0 µ =3q 0 (F − 1) − p 0 , κ 0 p =(3− 5F )q 0 + p 0 , F =1+b √ tx, (23.42)p 0 ≡ (3 − 2F )F/a 2 , q 0 ≡ (F − 1)(x 2 − t 2 )/4a 2 x 2 t 2 , a, b const.