A New Integral Equation Form of Integrable Reductions of the Einstein Equations

We further develop the monodromy transformation method for analyzing hyperbolic and elliptic integrable reductions of the Einstein equations. The compatibility conditions for alternative representations of solutions of the associated linear systems with a spectral parameter in terms of a pair of dressing (scattering) matrices yield a new set of linear (quasi-Fredholm) integral equations that are equivalent to the symmetry-reduced Einstein equations. In contrast to the previously derived singular integral equations constructed using conserved (nonevolving) monodromy data for fundamental solutions of the associated linear systems, the scalar kernels of the new equations involve functional parameters of a different type, the evolving (dynamic) monodromy data for scattering matrices. In the context of the Goursat problem, these data are completely determined for hyperbolic reductions by the characteristic initial data for the fields. The field components are expressed in quadratures in terms of solutions of the new integral equations.