Abstract <p>In this work, a vector three-dimensional inverse problem of diffraction by a cylindrical body is solved based on a two-step method. The scatterer is filled with an inhomogeneous nonmagnetic dielectric material. The initial boundary value problem for the system of Maxwell’s equations is reduced to a system of integro-differential equations. A numerical method for solving the equation of the first-kind, entering into the system, in special classes of functions is described. A distinctive feature of the proposed numerical method is its non-iterative nature; in addition, a good initial approximation is not required for the two-step method for solving the inverse problem. The calculation results are presented. It is shown that the two-step method is an efficient approach to solving vector problems of microwave tomography.</p>