Monodromy Transform Approach to Solution of Some Field Equations in General Relativity and String Theory

A monodromy transform approach, presented in this communication, provides a general base for solution of space-time symmetry reductions of Einstein equations in all known integrable cases, which include vacuum, electrovacuum, massless Weyl spinor field and stiff fluid, as well as some string theory induced gravity models. There were found a special finite set of functional parameters which are defined as the set of monodromy data for the fundamental solution of associated spectral problem. These monodromy data consist of the functions of the spectral parameter only. Similarly to the scattering data in the inverse scattering transform, the monodromy data can be used for characterization of any local solution of the field equations. A ”direct” and ”inverse” problems of such monodromy data transform admit unambiguous solutions. For the linear singular integral equation with a scalar (i.e. non-matrix) kernel, which solves the inverse problem of this monodromy transform, an equivalent regularization – a Fredholm linear integral equation of the second kind is constrcuted in several convenient forms. The existence and uniqueness of the local solution for arbitrary choice of the monodromy data can be proved using a simple iterative method. This solution is effectively constructed in terms of homogeneously convergent functional series.