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532 34 Application of generation techniques to general relativitythe basic generators A 1 , ..., A 6 are given byA 1 = −X 0 + Y 1 , A 2 = X 0 + Z 1 , A 3 = 1 2 (V 2 − V 0 ),A 4 = −X 0 + Y −1 ,A 5 = X 0 + Z −1 ,A 6 = 1 2 (V 0 − V −2 ),(34.45)cp. Guo et al. (1982). The loop algebra of SU(1,1) can be representedby setting X i = λ i X 0 etc., thereby introducing the spectral parameterλ. However, due to the presence of the Virasoro algebra this spectralparameter cannot be constant, rather it is another pseudopotential. Usinga one-dimensional non-linear representation for SU(1,1), viz.X 0 = y∂ y , Y 0 = y 2 ∂ y , Z 0 = ∂ y , (34.46)we find for (34.41) with two pseudopotentials y and λdy =[(λy 2 − y)M 3 +(λ − y)M 4 ]dζ+[(y 2 λ −1 )M ∗3 +(λ −1 + y)M ∗4 ]dζ,dλ = 1 2 λ(λ2 − 1)M 2 + 1 2 (λ − λ−1 )M 2 .(34.47)In terms of a given solution M i ,M ∗i and the pseudopotentials y and λcalculated from it, a new solution is given by a linear ansatz, viz.˜M 2 = u(λ)M 2 , ˜M ∗2 = v(λ)M 2 ,(34.48)˜M 3 = f i M i , ˜M 4 = g i M i , ˜M ∗3 = fi ∗ M ∗i , ˜M ∗4 = gi ∗ M ∗i .In the course of the ensuing calculations one finds that u = v −1 equalseither λ 2 or 1. The original choice in Harrison (1978) was u = v = 1. Thesix functions f i (y, λ) etc. can be determined algebraically. Harrison (1983)has given an exhaustive list. He has also analysed the group structure ofthe transformations. We just quote one example:˜M 2 = λ 2 M 2 , ˜M 3 = yλM 3 , ˜M 4 = y −1 λM 4˜M ∗2 = λ −2 M ∗2 , ˜M ∗3 = yλ −1 M ∗3 , ˜M(34.49)∗4 = y −1 λ −1 M ∗4(Neugebauer 1979, Kramer and Neugebauer 1981, 1984).34.3 The linearized equations, the Kinnersley–Chitre B groupand the Hoenselaers–Kinnersley–Xanthopoulostransformations34.3.1 The field equationsIn this section we shall derive a linear problem for the stationary axisymmetricvacuum equations of the form (10.58) from first principles