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№4 | 2021 32 of B¨ acklund transformations and other features in the group-theoretic context. Namely, a construction of B¨ acklund transformations (for vacuum time-dependent as well as for stationary axisymmetric fields) were described by Harrison [21]2, who used the pseudopotential method of Wahlquist and Estabrook [22]. In [21], the corresponding equations were expressed in terms of closed ideal of differential 1-forms. Later, Harrison described particular applications of B¨acklund transformations and generalized his approach to electrovacuum fields. A bit later, Neugebauer [25] presented his form of B¨acklund transformations for vacuum Einstein equations for stationary axisymmetric fields, which were constructed in the spirit of the known theory of B¨acklund transformations for sine-(sinh-)Gordon equation. In a short series of papers [25]-[27], Neugebauer found compact (determinant) form of N-fold B¨acklund transformations expressed in terms of the Ernst potential of some chosen beginning (or “initial”, or “background”) solution and of the solutions of Riccati equations with coefficients depending on this choice. Exponentiating of some of Kinnersley-Chitre infinitesimal symmetries: HKX-transformations. Another solution generating method for vacuum fields had been found by C.Hoenselaers, W.Kinnersley and B.C.Xanthopoulos. This method, called later as HKX- transformations and described in [28, 29], was derived as a result of exponentiation of some kinds of infinitesimal Kinnersley-Chitre transformations. The corresponding “rank p” transformations allow to obtain from a given initial vacuum stationary axisymmetric solution a new family of solutions of this type with p+ 1 arbitrary real parameter. The authors suggested also a construction of superposition of such transformations with different parameters and gave the simplest examples. Hauser and Ernst homogeneous Hilbert problem (HHP) and their integral equation method for effecting Kinnersley-Chitre transformations. Hauser and Ernst suggested yet another approach to generation of stationary axisymmetric vacuum [30], [32], [34], [35] and electrovacuum solutions. Within the class of solutions which are regular in some neighborhood of at least one point of the symmetry axis, the problem of “effecting” (i.e. exponentiating) of the Kinnersley and Chitre infinite-dimensional algebra of infinitesimal symmetry transformations was reduced to solution of a homogeneous Hilbert problem (HHP) on a closed contour on the plane of auxiliary complex parameter, which was reduced then to solution of a matrix linear singular integral equation of the Cauchy type on this contour. Solution of this integral equation for any chosen initial (“seed”) solution and arbitrarily selected element of Kinnersley-Chitre algebra, having been found, allows to calculate explicitly the transformed solution. However, the suggested in rational ansatz with too simple algebraic structure for solving this integral equation, had led to the class of solutions essentially more restricted in the number of free parameters than the class of electrovacuum solitons. Two years later, using the already mentioned above pseudopotential method of Wahlquist and Estabrook, Kramer and Neugebauer constructed for another set of pseudopotentials4 a linear system which integral condition is also provided by Einstein - Maxwell equations for stationary axisymmetric electrovacuum fields and then, Neugebauer and Kramer translated into the context of the system a constructions of soliton solutions in the spirit of inverse scattering transform, which was used earlier in for vacuum and in for electrovacuum fields. Nonetheless, in addition to the results oF, where the calculation of all components of metric (besides only the conformal factor) and of electromagnetic potential for electrovacuum soliton solutions have been constructed6), a useful input from the paper was a derivation of compact determinant expressions for the Ernst potentials for stationary axisymmetric electrovacuum solitons. The construction of B¨acklund transformations for electrovacuum Ernst equations have been described by Harrison. An interesting feature in this paper is an application of a modified Wahlquist-Estabrook approach, suitable for systems of equations, which can be expressed in terms of differential forms which constitute a closed ideal with constant coefficients (CC-ideal). Many known integral systems can be cast into such form. For these cases, it is possible to formulate simple general ansatzes which lead to a construction in some unified form of associated linear |