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A cylindrical structure composed of a circular, PEC cylinder, having a longitudinal slot and enclosing an arbitrary number of eccentric, dielectric cylinders, embedded in a dielectric sheath, is considered in this paper. Excitation is provided by a plane wave, normally incident upon the structure. Both TE and TM polarizations are considered for the incident wave. The scattered wave, as well as the electromagnetic (EM) field within the scatterer, are determined by using a dual-series approach combined with the translational addition theorem for cylindrical wave functions. The Abel integral-transform method is applied in order to regularize the dual-series equations to a matrix Fredholm equation of the second kind. The resulting infinite set of linear algebraic equations (i.s.l.a.e.) is truncated and solved numerically. The solution inherently contains the behavior of the field near the aperture rim, as required by Meixner's edge condition. The numerical application manifests how the internal structure of the scatterer and the polarization of the incident wave affect backscattering by the cylindrical cavity.