In this paper we consider the Cauchy problem for systems of integral-differential equations with zero points of the spectrum of the limit operator. The presence of singularities in the integral term with a rapidly decreasing kernel generates in a solution of the original problem of essentially special singularities by a small parameter, which describes the internal boundary layer. To establish mathematical theory, criterion formulation of correctness of mathematical description of the boundary layer and to develop a regular theory for singularly perturbed problems we use the regularization method of S.A.Lomov. Normal and unique solvability of iterative tasks is proved, the asymptotic convergence of formal solutions is proved.