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512 33 Algebraically special perfect fluid solutionswhich is, therefore, expansion- and shear-free (rigid motion). Only Petrovtypes II and D can occur; the metrics of type D are subcases or limitsof the Wahlquist solution (21.57).Special solutions (of Petrov type D) areK + κ 0 m =0 ⇒ W =0, R = R(¯ζ), (33.34)which for R = const contains a solution admitting a G 4 on r = const, cp.§13.4, and the solution (Kramer 1984c)[]K = −κ 0 m w(ζ + ζ)+3 , P −2 =e w+2 w/κ 0 m, w ′ = we 1+w/2√ 2,(33.35)see also (33.13), where the same functions appear in a different solution.33.3 Type D solutionsMost of the known algebraically special perfect fluid solutions are ofPetrov type D, but, as the following review shows, only a minor subset ofall such type D solutions is known.In the type D solutions, the two multiple null eigenvectors l and kdefine a preferred 2-space Σ. The four-velocity u may or may not lieinΣ.For u [a k b l c] ̸= 0, most of the known solutions admit an AbelianG 2 ofmotions, cp. Chapters 21 and 23, and many correspond to the interior ofa rigidly rotating body. Examples with a G 2 are the Wahlquist solution(21.57) and its limits, see §21.2.3, the solutions due to Mars and Senovilla(1994) and Mars and Wolf (1997) (admitting an additional conformalmotion), and a solution belonging to the Kerr–Schild class (Martín andSenovilla 1986), see §32.5.4. An example of a solution with only a G 1(but two conformal symmetries) is the metric (35.78) due to Koutras andMars (1997).For u [a k b l c] =0, (33.36)all known solutions have the property that the rotation ω a = ε abcd u b u c;dand shear σ ab of the velocity field furthermore obeyω [a k b l c] =0, k d σ d[a k b l c] =0. (33.37)Following Wainwright (1977b), we classify all solutions satisfying (33.36)and (33.37) according to the acceleration ˙u and the Newman–Penrosecoefficients κ, ν, σ and λ (it can be shown that for the solutions in questionκ =0(σ = 0) if and only if ν =0(λ = 0)). For none of the subcases isthe complete list of solutions known.