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For the fields depending on two of the four space-time coordinates only, thespaces of local solutions of various integrable reductions of Einstein's fieldequations are shown to be the subspaces of the spaces of local solutions of the``null-curvature'' equations constricted by a requirement of a universal (i.e.solution independent) structures of the canonical Jordan forms of the unknownmatrix variables. These spaces of solutions of the ``null-curvature'' equationscan be parametrized by a finite sets of free functional parameters -- arbitraryholomorphic (in some local domains) functions of the spectral parameter whichcan be interpreted as the monodromy data on the spectral plane of thefundamental solutions of associated linear systems. Direct and inverse problemsof such mapping (``monodromy transform''), i.e. the problem of finding of themonodromy data for any local solution of the ``null-curvature'' equations withgiven canonical forms, as well as the existence and uniqueness of such solutionfor arbitrarily chosen monodromy data are shown to be solvable unambiguously.The linear singular integral equations solving the inverse problems and theexplicit forms of the monodromy data corresponding to the spaces of solutionsof the symmetry reduced Einstein's field equations are derived.Comment: LaTeX, 33 pages, 1 figure. Typos, language and reference correction

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