Monodromy-data parameterization of spaces of local solutions of integrable reductions of Einstein's field equations

We show that for the fields depending on only two of the four space-time coordinates, the spaces of local solutions of various integrable reductions of Einstein's field equations are the subspaces of the spaces of local solutions of the “null-curvature” equations selected by universal (i.e., solution-independent conditions imposed on the canonical (Jordan) forms of the desired matrix variables. Each of these spaces of solutions can be parameterized by a finite set of holomorphic functions of the spectral parameter, which can be interpreted as a complete set of the monodromy data on the spectral plane of the fundamental solutions of associated linear systems. We show that both the direct and inverse problems of such a map, i.e., the problem of finding the monodromy data for any local solution of the null-curvature equations for the given Jordan forms and also of proving the existence and uniqueness of such a solution for arbitrary monodromy data, can be solved unambiguously (the “monodromy transform”). We derive the linear singular integral equations solving the inverse problem and determine the explicit forms of the monodromy data corresponding to the spaces of solutions of Einstein's field equations.