Discovery of integrability of symmetry reduced vacuum Einstein equations and formulation of the well known now Belinski-Zakharov inverse scattering approach to solution of these equations (including the soliton generating transformations and reformulation of the problem for ``non-soliton'' part of the solutions in terms of some matrix Riemann-Hilbert problem) more than thirty years ago opened the ways for construction in General Relativity of many physically interesting solutions as well as for development of similar (or based on the same basic ideas and appropriately modified) approaches to solution of other (non-vacuum) integrable symmetry reductions of Einsteins field equations (the Einstein-Maxwell and Einstein-Maxwell-Weyl equations in General Relativity, the bosonic field equations in some string gravity and supergravity models in four and higher dimensions and some others).
In this talk, we recall a general construction of some form of the linear integral equations found later by the present author and available for solution of any known today integrable reductions of Einstein's field equations. This form of (quasi-Fredholm) linear integral equations is most appropriate for solution of the characteristic initial value problems for these equations. An application of this integral equation method to construction of solution for nonlinear gravitational interaction of short electromagnetic pulses colliding on the expanding cosmological background is considered.