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The monograph is devoted to the study of non-correct problems for a system of Maxwell-type equations with an elliptic complex in a bounded region. We consider the Cauchy problem for an elliptic system of equations of Maxwell type in "n" dimensional space. And also the construction of the matrix of the left fundamental solution of the system of electrodynamics equations of a special kind is given. The generalized Stratton-Chu formula is obtained here, the solvability conditions are proved and the Carleman formula is constructed using the method of bases with double orthogonality, i.e. an explicit formula is constructed which restores the solution of the system of Maxwell's equations in a bounded region.