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№4 | 2021 33 systems (and corresponding “spectral problems”) as well as of B¨acklund transformations for these systems. More later, a spectral problem of yet another structure for the same stationary axisymmetric electrovacuum Einstein - Maxwell field equations in the form of a sigma-model was suggested in [40]. In this more geometrical context, calculation of soliton solutions, also was performed in the spirit of the inverse scattering approach, but the used form of the spectral problem has led directly to calculation of the corresponding Ernst potentials only. On the relations between the solution generating methods. Close interrelations between different approaches to construction of vacuum solution generating methods (the inverse scattering method, theory of B¨acklund transformations and group-theoretical approach) were described by Cosgrove [41] – [43]. The relations between associated linear systems (“spectral problems”) used by different authors for generalizations of their approaches to electrovacuum fields was found by Kramer. On the difficulties with explicit applications of solution generating methods. The general studies of the families of solutions generating with the methods mentioned above occur rather difficult because these families of solutions do not admit their representation in general and explicit form, due to a presence in these solutions, besides a large number of constant parameters, some functional parameters – the potentials which characterize the chosen initial (background) solution. In each of these methods, these potentials should satisfy some linear systems with coefficients depending on the choice of initial solution, but this system can be solved explicitly not for any choice of the initial solution. Only in those cases, in which for chosen initial (background) solution this linear system can be solved explicitly, one can calculate all components of the solutions generating on this background. On the “coordinates” in the space of solutions. The difficulties mentioned just above can be overcome if we introduce in the space of solutions, instead of metric and field components, some “coordinates” which, from one hand, would be related to various physical and geometrical characteristics of the solutions and, from the other hand, different solution generating procedures could be represented as transformations of these “coordinates”. In the most general cases, for G2-symmetry reduced vacuum Einstein equations and electrovacuum Einstein - Maxwell equations, the role of such “coordinates” in the infinite- dimensional spaces of their local solutions can belong to the monodromy data of the fundamental solutions of the corresponding associated linear systems (“spectral problems”). However, for large subclasses of field configurations which possess the asymptotic behaviour of the same type near some space-time boundaries, such “coordinates” can be defined in a simpler way. For example, for stationary axisymmetric fields the role of such “coordinates can be played by the values of the Ernst potentials on those parts of axis of symmetry near which the space-time geometry and electromagnetic fields possess regular behavior. In this paper, we consider the classes of fields, which possess, similarly to the regular parts of axis of symmetry in axisymmetric fields, the boundaries consisting of degenerate orbits of the isometry group G2 with regular behavior of metric and electromagnetic fields near these boundaries (see below for more details). It is clear that besides the stationary axisymmetric fields near the regular parts of the axis of symmetry, these classes of fields include, in particular, cylindrical waves and some other types of wave-like or cosmological solutions, stationary fields with Killing horizons as well as some other types of solutions which can have, in particular, a dynamical nature (like the well-known “C-metrics”). For all these types of fields, the “coordinates” in the space of solutions may be represented by the functional parameters defined as the values of the Ernst potentials on the lines in the orbit space which consist of the points (orbits) at which the geometry of the orbits is degenerate, but the space-time geometry remains regular. These “coordinates” determine the corresponding local solutions “almost unequally”. As it will be shown further, different solution generating procedures can bepresented explicitly and in a very simple form as transformations of the described above “coordinates” in the spaces of local solutions. It is very important that these transformations possess a general form which does not need to specify in advance the choice of the initial solution. The initial solution in |