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№4 | 2021 31 arbitrarily chosen known solutions equations, to construct infinite hierarchies of solutions with an arbitrary (finite) number of free parameters. Group-theoretic approach : Geroch group and Kinnersley-Chitre algebra. The development of the group-theoretic approach to the studies of internal symmetries of vacuum Einstein equations and electrovacuum Einstein – Maxwell equations began long ago from beautiful discoveries of symmetry transformations space-times with at least one Killing vector field. In the previous century, these transformations were found at the end of 50 th for vacuum by J.Ehlers and at the end of 60th for electrovacuum by B.K.Harrison. Later, it was found that these transformations represent the subgroups in a larger group of symmetries isomorphic respectively for vacuum and electrovacuum. At the beginning of 70th, R.Geroch published a paper where he conjectured that for vacuum space-times with two commuting non-null Killing vector fields for which two certain real constants vanish (these conditions are equivalent to the mentioned above G2-symmetry conditions), there exists an infinite dimensional group of internal symmetries which action on the space of solutions is transitive, i.e. every solution can be obtained applying the symmetry transformation to a chosen solution, e.g., to the Minkowski space-time. Geroch also argued that the corresponding infinite- dimensional algebra of infinitesimal transformations can be built inductively. Later, W.Kinnersley extended these considerations to the case of stationary 2axisymmetric electrovacuum fields.1 In the subsequent papers, W.Kinnersley and D.M.Chitre presented a systematic study of the infinite-dimensional algebra of infinitesimal symmetries of Einstein - Maxwell equations for stationary axisymmetric fields. In these papers, an infinite hierarchies of complex matrix potentials associated with every particular solution were constructed, and it was shown that these hierarchies of potentials form a representation space of this algebra. Moreover, it was found that for electrovacuum case, these sets of potentials admit two 3 × 3 (or 2 × 2 in vacuum case) matrix generating functions, one of which happen to satisfy a linear system of equations with an auxiliary complex parameter, and another generating function can be expressed algebraically in terms of the first one. Some years later, in the papers of B.Julia [13, 14] the infinite dimensional symmetry transformation of Geroch and Kinnersley and Chitre were recognized as Kac-Moody symmetries and in the paper of P.Breitenlohner and D.Maison [15] the structure of the corresponding infinite- dimensional Geroch group was described in detail. However, at that time, the problem of exponentiating of Kinnersley and Chitre infinitesinal transformations for obtaining new solutions of Einstein - Maxwell equations had remained to be solved. Soliton solutions of Einstein and Einstein - Maxwell field equations. Using very different approach based on the ideas and methods of the inverse scattering theory, in the pioneer papers [16, 17], V.A.Belinski and V.E.Zakharov discovered the existence of infinite hierarchies of exact Nsoliton solutions of vacuum Einstein equations depending on 4N free real parameters. These solitons can be generated on arbitrarily chosen vacuum background with the mentioned above G2-symmetry. It is important also that in these papers the explicit expressions had been obtained for all metric components, including the so called conformal factor – the coefficient in front of conformally flat part of metric. This factor was expressed in [16, 17] explicitly in terms of the components of chosen vacuum background metric and solution of the corresponding spectral problem. More compact, determinant form of Belinski and Zakharov N-soliton solutions was found in the author’s paper [18]. It is worth mentioning also that in [16, 17], a 2 × 2-matrix linear singular integral equation with the kernel of a Cauchy type, was constructed for generating “non-soliton” vacuum solutions. A bit later, in the author’s papers [19, 20], using the same general ideas and methods of the inverse scattering approach (but for essentially differennt, complex self-dual form of Einstein- Maxwell equations found by W.Kinnersley [9]), the N-soliton solutions of Einstein - Maxwell equations depending on 3N free complex or 6N real parameters were constructed starting from arbitrarily chosen (G2 symmetric) electrovacuum background. Backlund transformations. Some other solution generating methods suggested later for G2- symmetry-reduced vacuum Einstein equations were constructed using the basic ideas of the theory |