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A 3-D extension of the adaptive integral method (AIM) is presented for fast analysis of scattering from electrically large perfect electrically conducting structures embedded inside a single layer of a planar layered medium. The proposed scheme accelerates the iterative method-of-moments (MOM) solution of the combined-field integral equation by employing a 3-D auxiliary regular grid. It uses the auxiliary grid to execute the standard four-stage AIM procedure; unlike the procedure for free space, two different sets of matrices are obtained for the AIM propagation stage by decomposing the Green functions to terms that are in convolution or correlation form in the stratification direction. These matrices are in (three level) block-Toeplitz and Hankel-(two level)-block-Toeplitz forms and can be multiplied by using 3-D FFTs. The dominant computational costs of the scheme are the evaluation of O(N) different layered-medium Green functions, which is accelerated by extracting asymptotic terms and using interpolation tables, and the matrix multiplications in the propagation stage, which require only O(NC log NC) per iteration; these should be contrasted to the O(N2) Green function evaluations and O(N2) operations per iteration required by the classical MOM. Here, NC denotes the number of nodes on the auxiliary grid, and N denotes the number of degrees of freedom of the surface current density. Numerical results validate the proposed method's complexity, demonstrate its accuracy for several large-scale structures in layered media, and compare its computational costs to those of its counterpart for free space.