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Table 11.2. Solutions with proper homothety groups Hr, r>4The H4 act on V4 except in (24.46) where the action is on N3. References are to where the homothety (not thesolution) was first given. The proper homothety generator is only defined up to addition of a Killing vector. Theenergy-momentum notation is defined in the main text.Hr Metric Tab Homothety Reference(s)H8 (12.37) [in (12.12)] E 2v∂v + ζ∂ζ + ¯ζ∂¯ζ Eardley (1974)H7 (12.12), (37.104) E 2v∂v + ζ∂ζ + ¯ζ∂¯ζ Eardley (1974)(14.8b) F t∂t +(1− 2/3γ)r∂r Eardley (1974)§14.2: (12.9), ε ̸= 0,γ = 2 3F t∂tH6 (24.46) E 2v∂v + ζ∂ζ + ¯ζ∂¯ζ Eardley (1974)H5 LRS subcase of (13.51)≡(22.7), cp. V x 4 ∂ x 4 + ∑ 3i=1 (1 + a4 − ai)x i ∂ x i Godfrey (1972)(13.53) and Table 18.2, including(15.12), k =0,e 2ν =1/x 3 (BIII) V 4x 4 ∂ x 4 +2x 3 ∂ x 3 + x 1 ∂ x 1 Koutras (1992b)(15.17), k =0,H = −2m/r (AIII) V 4u∂u +2r∂r + x 1 ∂ x 1 Halford and Kerr (1980)(13.79) F w∂w Daishev (1984)(13.85), a =0 F 4x∂x +(2q + 1)(y∂y + z∂z + t∂t) Wainwright (1985)(14.19) F t∂t +(1− c)x∂x Sintes (1996)(14.20) F (13.52), a4 =0,ai = pi Hsu and Wainwright (1986)(14.28b), LRS case F (13.52) Koutras (1992b)(15.17), k =0,H = e 2 /r 2 E 3u∂u + r∂r + x 1 ∂ x 1 Carminati and McIntosh (1980)(15.50), L = 1: see §15.6.2 F r∂r + at∂t(15.65) with A = a/r n ,B= br n F t∂t + r∂r Ponce de León (1988)(15.76a) with 6/7