Integrable structure of the low-energy string gravity equations in D=4 space-times with two commuting isometries

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.