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140 10 Generation techniques10.4.2 Isovectors, similarity solutions and conservation lawsIn what follows we shall assume that we are given a closed ideal of differentialforms. A characteristic vector is defined by the property thatcontracting it with any form of the ideal gives again a form in the ideal;it corresponds to a ‘conditional symmetry’. For instance, let the set inquestion consist of 1-forms ω A and 2-forms α a . A characteristic vectorsatisfiesφ a AY ω A =0, Y α a = φ a A ω A =0(mod ideal) : (10.46)are undetermined functions. If there is another characteristic vector,Z, we find by taking the Lie derivative of (10.46) with respect to Z that[Y, Z] is also a characteristic vector. Thus the characteristics are surfaceforming.We note that adjoining a characteristic to each point of a regularintegral manifold gives again a regular integral manifold. The maximalintegral manifold thus must contain all characteristics.A generalization of a characteristic vector is an isovector. Now it is theLie derivative of any form in the ideal which has to be in the ideal. Thus,for the example considered above, X is an isovector ifL X ω A =0(mod ideal), L X α a =0(mod ideal). (10.47)X is a solution of an overdetermined set of linear partial differential equations;any constant linear combination of isovectors is again an isovector.Moreover, if X and Y are isovectors then [X, Y] is again an isovector.The set of all isovectors generates the isogroup. Isovectors correspond toLie–Bäcklund symmetries, cp. §10.3.1.An ideal of forms remains invariant under the infinitesimal mappingsgenerated by its isovectors. Thus the isovectors also map integral manifoldsinto integral manifolds and the finite transformations, found inanalogy to (10.87), generate new solutions of the underlying differentialequations.Given an isovector which is not a characteristic vector a number of newforms can be found by contracting it with the forms in the ideal. Adjoiningthese to the original ideal one finds a larger closed ideal of forms theintegral manifold of which is a subset of the integral manifold of the originalideal. These are the most general similarity solutions. The integrationto find similarity solutions involves at least one less independent variablesince the larger ideal has a characteristic vector.Suppose that, in a given closed ideal of genus g, we have found an exactk-form, k =1,...,g. Denote this exact form by dϑ, where ϑ isa(k − 1)-form determined up to the exterior derivative of a (k − 2)-form. From