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336 21 Non-empty stationary axisymmetric solutionsstate is µ = p + const and the line element is given by[ds 2 = N −2 (z) dz 2 + G −1 (x)dx 2 + G(x)dϕ 2 − x 2 (dt + nx −2 dϕ) 2] ,G(x) =εx 2 + b ln x + c + n 2 /x 2 , N ′′ (z) =εN(z), (21.61)κ 0 (p − µ) =6(N ′2 − εN 2 ), κ 0 (µ + p) =bN 2 (z)/x 2 , ε =1, 0, −1.This last metric generalizes the static solution (18.66) to which it reducesfor n = 0. The solutions with µ = p have flat three-dimensional slices (see(4.9) of Wolf (1986b)) and admit the homothetic vector H = ∂ z (see alsoHermann (1983)).Static solutions admitting an additional homothetic vector have beenfound by Kolassis and Griffiths (1996).Using the field equations as given by Bonanos and Sklavenites (1985),a solution of Petrov type I, with a vanishing magnetic part of theWeyl tensor, and an equation of state µ = p + const has been found bySklavenites (1985) and Kyriakopoulos (1999).Starting with special assumptions for the metric functions, rigidly rotatingsolutions have been found by Kyriakopoulos (1987, 1988) and Sklavenites(1992b). The solutions (36.34)–(36.35) are also rigidly rotating.Besides the two Killing vectors, a rigidly rotating perfect fluid mayadmit an additional proper conformal vector (Kramer 1990, Kramer andCarot 1991). For the only known non-static solution with this propertythe three symmetries commute, and the metric is given by (21.61).Known rigidly rotating axisymmetric perfect fluid solutions with morethan two Killing vectors belong to the locally rotationally-symmetricspace-times (§§13.4, 14.3) or to the homogeneous space-times (§12.4) orthey are cylindrically-symmetric (§22.2). For instance, if in generalizingPapapetrou’s class of vacuum solutions (§20.3) one assumes ω = ω(U),ω ,ζ = −iW −1 e 4U A ,ζ and µ +3p ̸= 0 (Herlt 1972), then one arrives at alocally rotationally-symmetric metric (13.2).The solutions due to Wahlquist and Herlt, and the interior Schwarzschildsolution (16.18) are the only perfect fluid solutions (with rigid rotation)for which the Hamilton–Jacobi equation for null geodesics is separable(Bonanos 1976). The only stationary, axisymmetric and conformallyflat perfect fluid solution is the interior Schwarzschild solution (16.18)(Collinson 1976b).Jackson (1970) applied the ‘complex trick’ (§21.1.4) to sphericallysymmetricperfect fluids, in particular to the interior Schwarzschild solution,and obtained an interior NUT metric. Herrera and Jimenez (1982)and Drake and Szekeres (1998) discussed also a generalization which appliesthe method to spherically-symmetric metrics, including fluid cases,