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34.5 The Belinski–Zakharov technique 5431985a). Ernst (see Hauser and Ernst 1979a) has used data specified on thesymmetry axis to determine the parameters K i of Neugebauer’s Bäcklundtransformation.34.5 The Belinski–Zakharov techniqueIn this section we shall introduce the main features of the Belinski–Zakharovtechnique (Belinskii and Zakharov 1978, 1979); for reviews, see Miccichè(1999) and Belinski and Verdaguer (2001). Belinski and Zakharovstarted with the matrix g representing the ϕ–t-part of the metric, i.e.( )−1 −Ag = W e −U , (34.97)−A e 2U + A 2and introduced the matricesA = −W g ζ g −1 , B = W g ζg −1 (34.98)(ζ given by (34.77)). The important part of the field equations, i.e. thepart involving the ϕ–t-components of the metric, becomesA ζ− B ζ =0. (34.99)Apart from this equation, there are integrability conditions following from(34.98). They areW (A ζ+ B ζ )+[A, B] − W ζA − W ζ B =0. (34.100)We are thus interested in two matrices A and B satisfying (34.99) and(34.100).Belinski and Zakharov introduced the following operatorsD 1 = ∂ ζ −2λλ − W W ζ∂ λ , D 2 = ∂ ζ + 2λλ + W W ζ ∂ λ. (34.101)It can be shown by direct calculation that the condition that the Dscommute is equivalent to the equation for W , i.e.[D 1 , D 2 ]=0 ⇐⇒ W ζζ=0. (34.102)Moreover, the Ds are invariant under the rescalingλ −→ λ ′ = W 2 λ −1 . (34.103)A linear pair, analogous to (10.58) albeit using two derivative operatorsin place of the exterior derivative, is introduced byD 1 Φ=1λ − W AΦ, D 2Φ=1BΦ, (34.104)λ + W