  A unified general approach is presented for construction of solutions of the
characteristic initial value problems for various integrable hyperbolic
reductions of Einstein's equations for space-times with two commuting
isometries in General Relativity and in some string theory induced gravity
models. In all cases the associated linear systems of similar structures are
used, and their fundamental solutions admit an alternative representations by
two ``scattering'' matrices of a simple analytical structures on the spectral
plane. The condition of equivalence of these representations leads to the
linear ``integral evolution equations'' whose scalar kernels and right hand
sides are determined completely by the initial data for the fields specified on
the two initial characteristics. If the initial data for the fields are given,
all field components of the corresponding solution can be expressed in
quadratures in terms of a unique solution of these quasi - Fredholm integral
evolution equations.
