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24.4 Null Killingvectors (G 1 on N 1 ) 379These Killing vectors necessarily commute. The metric (24.18) is independentof x, and the rest of the Killing equations imply W ,v ̸= 0. Thetype D vacuum solution (31.41) is also contained hereThe general Einstein–Maxwell type D solutions with a G 2 on null orbits,where both eigenvectors of the Weyl and Maxwell tensor are aligned andgeodesic and shearfree, divide into two classes. They have either one nulleigenvector with non-zero expansion and twist and are then given byds 2 = (x2 + y 2 )dx 22nx − (e 2 + g 2 ) + 2nx − (e2 + g 2 )x 2 + y 2 [du − y 2 dv] 2 ,+2dy[du + x 2 dv],Φ 11 = −(e 2 + g 2 )/2(x 2 + y 2 ) 2 ,(24.21)or both null eigenvectors are expansion- and twist-free and the solutionsare given byds 2 = K(x)x 2 + l 2 [dv +2lydu]2 + x2 + l 2K(x) dx2 +2(x 2 + l 2 )dydu,(K(x) =2nx − (e 2 + g 2 ) − Λ 2l 2 x 2 + x 4 /3 − l 4) , (24.22)Φ 11 = −(e 2 + g 2 )/2(x 2 + l 2 )(García D. and Salazar I. 1983). If admissible, the cosmological constant Λhas been included. Solution (24.21) is a special case of the solutions givenby Leroy (1978) and Debever (1971). Solution (24.22) contains the vacuumsolutions found by Bampi and Cianci (1979), see also Joly et al. (1992).Both classes can be obtained by limiting procedures (contractions) fromthe type D solutions with a G 2 on non-null orbits (García D. and SalazarI. 1983, García D. and Plebański1982b), cp. §21.1.2.24.4 Null Killing vectors (G 1 on N 1 )The important relation (6.33),Θ ,a k a − ω 2 +Θ 2 + σ¯σ = − 1 2 R abk a k b , (24.23)applied to a null Killing vector k yieldsR ab k a k b =2ω 2 . (24.24)Obviously, for vacuum solutions, and for Einstein–Maxwell and pure radiationfields for which k is an eigenvector of R ab , the null Killing vector