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26.1 Solutions of Petrov type II, D, III or N 409where the real functions P (ζ,ζ,u) and B(ζ,ζ,u) have to satisfyP 2 ∂ ζ ∂ ζP 2 ∂ ζ ∂ ζln P − 3m(ln P ) ,u − m ,u − 2a 2 m 2 P −2+3amP −1 [P ,ζ B ,ζ+ P ,ζB ,ζ ] − P 2 B ,ζζ B ,ζ ζ=0, (26.5b)P 2 B ,ζζ= am.In the general case, it is of type II (and one can put a = 1 and m =1). For m = 0 it contains the type III metrics studied by Cahen andSpelkens (1967); here B is an arbitrary harmonic function. For B =0(⇒ m = 0) it contains the general non-aligned type N metrics found byCahen and Leroy (1965, 1966) and Szekeres (1966b). As given by Szekeres,it readsds 2 = 1 2 cos2 ar(dx 2 +dy 2 ) − 2b(2r + a −1 sin 2ar)du dx−4du dr +4a −2 (2b 2 sin 2 ar − 2e 2u − raa ,u )du 2 , (26.6)a(x, u) =g(u) cosech [e u x + f(u)] , b(x, u) =−e u coth [e u x + f(u)] ,with k n = u ,n . The limit a = 0 of (26.5) leads to the Robinson–Trautmanmetrics (Chapter 28).In the aligned case, (26.3) is satisfied by definition. As shown in §7.6,the Bianchi identities yieldσ(3Ψ 2 +2κ 0 Φ 1 Φ 1 )=0=κ(3Ψ 2 − 2κ 0 Φ 1 Φ 1 ), (26.7)so equation (26.2) can be violated only for Petrov type II (or D) solutionswith a special constant ratio between Ψ 2 and Φ 1 Φ 1 , and κ or σ must benon-zero (note that because of (6.33) κ = ρ = 0 would induce σ =0).The case κ =0,σ̸= 0, has been excluded by Kozarzewski(1965), soonly κ ̸= 0,σ =0, remains to be studied. If the two null eigenvectorsof a type D solution are aligned with the eigenvectors of the Maxwelltensor, then they must both be geodesic and shearfree; this is not true if acosmological constant Λ is admitted (see García D. and Plebański(1982a)and Plebański and Hacyan (1979), where also some solutions are given).For pure radiation fields, one again has to distinguish between thealigned and the non-aligned cases. So far, only the aligned case has beentreated (the general type N and type III metrics are necessarily aligned:Plebański(1972), Urbantke (1975), Wils (1989a)). Due to the theoremsgiven in §7.6, all aligned type II, D or III solutions have a geodesic andshearfree k. The only algebraically special, aligned, pure radiation fieldsnot covered by (26.1)–(26.2) are therefore of type N. Among the Weyl and