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34.5 The Belinski–Zakharov technique 545Finally, one requires det(g) =−W 2 . This implies for Pdet [P (λ = 0)] = 1. (34.114)In fact, any solution of the linear pair (34.104) is defined up to a scalingg → h(λ =0) −1 g. Thus, given any solution g of (34.105) the physicalmetric g ph (det g ph = −W 2 ) can be obtained byg ph = W (det g) 1/2 g, (34.115)in which case we have h = (det g) 1/2 W −1 .In order to construct a solution explicitly, we now introduce some assumptionson the pole structure of the matrix P in the complex λ-plane.Here, we shall assume that P and P −1 have singularities in λ and thatthese are simple poles, i.e. both P and P −1 are meromorphic. Of course,the poles will depend on W and V ; note that – whenever possible –we have suppressed the coordinate dependence of the various quantities.This is analogous to prescribing the zeros of the dressing matrix P in§34.4. Let us assume that P (λ) is not invertible at a number of points ν k(k =1,...,n) and that these are simple poles for P −1 . Then it can beshown that P has simple poles at µ k = W 2 νk −1 .From (34.108) it can be concluded that the poles of P are either real orcome in complex conjugate pairs. Thus the general forms of P and P −1aren∑]P =[(λ − µ k )R k +(λ − µ k ) −1 R k , (34.116a)P −1 =k=1n∑k=1[(λ − ν k ) −1 Q k +(λ − ν k )Q k], (34.116b)where the matrices R k and Q k are related through the condition PP −1 =1. By inserting (34.116) into (34.107) we find an equation for µ k thesolution of which is√µ k = w k ± (w k − V ) 2 + W 2 . (34.117)The R k cannot be chosen arbitrarily, rather they are given in terms ofvectors n k and m k byR k = n k m T k . (34.118)For a given seed metric g 0 one obtains, at least in principle, Φ 0 from(34.104). Then one defines matrices M k and with their help the vectorsm k byM k =Φ 0 (λ = µ k ), m k = M T k κ k , (34.119)