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35.4 Collineations and conformal motions 569all solutions when the V 2 is spacelike and the motions are an invariantsubgroup of a non-Abelian conformal group (and no extra symmetry ispresent). The (four) metrics are necessarily diagonal, cp. also §23.3.1, andone of them,ds 2 = t a x c−2 [dx 2 + t 1−a−b dy 2 + t 1−a+b dz 2 − dt 2 ],c =2a 2 /(b 2 + a 2 − 2a − 1) ̸= 0, a,b = const, (35.77)u n = −sign (c)t a/2 x c/2−1 [ a 2 x 2 − t 2] −1/2 (t, 0, 0, 2a/c) ,admits besides the conformal Killing vector ∂ x (which commutes with thetwo Killing vectors ∂ y and ∂ z ) a homothetic vector ζ =2t∂ t +2x∂ x +(1+a + b)y∂ y +(1+a − b)z∂ z . The related metric, with x and t interchangedin the metric functions, contains a metric found by Bray (1971). Czaporand Coley (1995) considered metrics with a spacelike V 2 and an inheritingconformal vector, cp. also Vera (1998a); the only perfect fluid diagonalspace-time with an equation of state p = p(µ) satisfying µ>0, µ+ p>0and admitting a proper conformal vector is the Allnutt solution m =0of(23.13). The case of null orbits (and a spacelike V 2 ) has been studied bySintes et al. (1998).For stationary axially-symmetric perfect fluids (i.e. with a G 2 actingon a timelike surface) with one additional conformal vector, the followingresults are known: If the resulting three-dimensional Lie algebra of ξ =∂/∂t, η = ∂/∂ϕ, and the conformal motion ζ is Abelian, then the solutions(of type D, equation of state p = µ + const, in general with differentialrotation) are (21.74) and a metric of Herlt’s class (the twist vector is agradient, see §21.2) with a conformally flat 3-space (Kramer 1992, 1990,Mars and Senovilla 1994). All non-static rigidly rotating solutions with aproper conformal vector are contained here (Kramer and Carot 1991).If for a non-rotating perfect fluid (with µ + p>0) there is an Abeliangroup of one Killing vector ∂ x and two conformal Killing vectors η and ζacting on a spacelike hypersurface, then in coordinates with η = ∂ y andζ = ∂ x the non-conformally flat solutions are of the formds 2 = N 1−α −1−αα (y)Q α (z)[A(t)dx 2 + t 1+α dy 2 + t 1−α dz 2 − A(t)dt 2 ],A(t) =a 0 + a 1 t 1−α + a 2 t 1+α , α (̸= 0),a i = const,(35.78)N ′2 = α 2 (a 1 N 2 − b 1 )q, Q ′2 = α 2 (a 2 Q 2 − b 2 ), b i = const,u a dx a = N 1−α −1−α2α (y)Q 2α (z)d(tN 1/α Q −1/α )(Koutras and Mars 1997). They are of Petrov type D and generalize thesolutions (32.98)–(32.99) found by Senovilla and Sopuerta (1994) using a