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This paper presents a generalization of the method of lines (MoL) numerical algorithm for the analysis of integrated structures with a finite local curvature. Starting from the derivation of the generalized transmission line equations in a generic orthogonal reference system and in a bianisotropic material, it is shown the possibility of solving the electromagnetic problem associated to an integrated structure with arbitrary curvature and/or loaded by complex dielectric materials via a MoL approach. The introduction of new algorithms that extend and complete the original method is required in the most general case. A detailed description of this new extended version of the MoL is presented through the paper and its full coincidence with the original algorithm in the case of structures with simple loading materials is also shown. Finally, some numerical results obtained analyzing microwave conformal antennas and resonators are presented and compared with analyzes performed employing other numerical methods and available in the literature.