  The generalized Einstein - Maxwell field equations which arise from a
truncated bosonic part of the low - energy string gravity effective action in
four dimensions (the so called Einstein-Maxwell - axion - dilaton theory) are
considered. The integrable structure of these field equations for D=4
space-times with two commuting isometries is elucidated. We express the
dynamical part of the reduced equations as integrability conditions of some
overdetermined $4\times 4$-matrix linear system with a spectral parameter. The
remaining part of the field equations are expressed as the conditions of
existence for this linear system of two $4\times 4$-matrix integrals of special
structures. This provides a convenient base for a generalization to these
equations of various solution generating methods developed earlier in General
Relativity.
