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246 15 Groups G 3 on non-null orbits V 2(comoving coordinates) admit a conformal vector ζ = N(z)∂ z , which ishomothetic for N = z. Contained here as k =0,N = z, B = sinh 2t, isametric due to Bray (1983) (with an equation of state µ = p), and as k =1, N= sinh az, B = sinh 2at, a solution due to Tariq and Tupper (1992).The metricsds 2 =dz 2 − dt 2 + P (z)Q(t)[dx 2 +dy 2 ],κ 0 (p − µ) =P ′′ /P − ¨Q/Q, κ 0 (p + µ) =(˙Q 2 /Q 2 − P ′2 /P 2 )/2,with P ′′ = ka 2 P, ¨Q = −ka 2 Q, k =0, ±1, (15.87)or P (z) = exp(bz 2 + c 1 z),Q(t) = exp(−bt 2 + c 2 t).are also perfect fluids (in non-comoving coordinates). Contained here asP (z) = exp z, Q(t) = cos t is a metric found by Bray (1983).Incompressible fluids (µ = const) have been investigated in Taub(1956). Davidson (1988) found the special solution (in comoving coordinates)ds 2 = tz m(m−1) [ dx 2 +dy 2] − 2dz dt − t −1 (z + z m t 1/(m−1) )dt 2 . (15.88)Götz (1988) considered metrics of the form[ds 2 = U(z)V (t) dx 2 +dy 2] + V (t) 1−α dz 2 − U(z) α dt 2 (15.89)with an equation of state p =(γ−1)µ; the equations for U and V decoupleand can be solved by quadratures.Following Taub (1972), Shikin (1979) determined implicitly all solutionsof the formds 2 = Y 2 (z/t)[dx 2 +dy 2 ]+X 2 (z/t)dz 2 − T 2 (z/t)dt 2 (15.90)(in comoving coordinates, and with an equation of state p =(γ − 1)µ)in terms of quadratures; they admit an additional homothetic vector ξ =x n ∂ n .Plane-symmetric solutions (in non-comoving coordinates) are containedin the metrics (36.20a) possessing flat slices. Plane-symmetric solutions ofembedding class one have been constructed by Gupta and Sharma (1996a,1996b). Metrics of plane symmetry also occur as subcases of perfect fluidsolutions admitting a G 2 on S 2 treated in Chapter 23.