  The integrability structures of the matrix generalizations of the Ernst
equation for Hermitian or complex symmetric $d\times d$-matrix Ernst potentials
are elucidated. These equations arise in the string theory as the equations of
motion for a truncated bosonic parts of the low-energy effective action
respectively for a dilaton and $d\times d$ - matrix of moduli fields or for a
string gravity model with a scalar (dilaton) field, U(1) gauge vector field and
an antisymmetric 3-form field, all depending on two space-time coordinates
only. We construct the corresponding spectral problems based on the
overdetermined $2d\times 2d$-linear systems with a spectral parameter and the
universal (i.e. solution independent) structures of the canonical Jordan forms
of their matrix coefficients. The additionally imposed conditions of existence
for each of these systems of two matrix integrals with appropriate symmetries
provide a specific (coset) structures of the related matrix variables. An
equivalence of these spectral problems to the original field equations is
proved and some approach for construction of multiparametric families of their
solutions is envisaged.
