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10.9 Variational Bäcklund transformations 151The equations for the other fields are obtained by the variational derivativeof the Lagrangian (10.95) with respect to the ϕ A . Replacing κ by ψin the field equations is tantamount to the Legendre transformationL ′ = L − κ ,a ∂L/∂κ ,a = G ′ ABϕ A ,aϕ B,a − ψ ,a ψ ,a /f(ϕ C ). (10.98)Clearly, L ′ admits at least one Killing vector, ∂ ψ , and the transformationassociated with it is ψ → ψ+ const which leaves κ unaffected. If, however,L ′ admits more than one Killing vector it can be used to generate newsolutions. Moreover, if this other Killing vector is hypersurface-orthogonalone can iterate the process and at each step generate new solutions.For a two-dimensional background space with real coordinates x andy, say, we can introduce derivative operators ∂ = ∂ x + i∂ y ,∂ ∗ = ∂ x − i∂ y ,respectively ∂ = ∂ x +∂ y ,∂ ∗ = ∂ x −∂ y , depending on whether the signatureis 2 or 0. Moreover, we use an n-bein in the potential space such thatG AB = e α A e Bα , G αβ = e A α e Aβ , (10.99)where we do not require the frame metric G αβ to be constant or diagonal.Projecting the derivatives of the fields ϕ A onto the frame vectors we defineM α = e α A ∂ϕ A , M ∗α = e α A ∂ ∗ ϕ A . (10.100)The field equations (10.84) together with the integrability conditions ofthese equations become∂ ∗ M α +Γ α βγM β M ∗γ =0, ∂M α +Γ α βγM ∗β M γ =0, (10.101)where Γ α βγ are the Ricci rotation coefficients, Γ α βγ = e αA e βA;B e B γ .The advantage of this formulation of the field equations is that the nonlinearitiesbecome purely quadratic if it is possible to choose the framevectors such that the rotation coefficients are constant. According to Theorem8.18 this is possible if G AB admits a simply-transitive n-dimensionalgroup of affine collineations.10.9Variational Bäcklund transformationsIn this section we assume that the equations in question are againEuler–Lagrange equations of the form (10.82) with a general LagrangianL(ϕ A ,ϕ A ,a,ϕ A ,ab,...) which is supposed to be polynomial in the derivativesof the fields. The field equations to be derived from this Lagrangian containderivatives of order 2n if L is of order n. IfL can be factorized,i.e.L = L 1 (ϕ A ,ϕ A ,a,ϕ A ,ab,...) L 2 (ϕ A ,ϕ A ,a,ϕ A ,ab,...) + divergence terms,(10.102)