arXiv:2610.29481v1 Announce Type: new Abstract: A symmetry-inheritance and exact domain-reduction framework is developed for the second-order vector wave equation of the electric field in bounded domains with a perfect electric conductor (PEC) boundary and general spatially varying anisotropic media. Orthogonal group actions on polar vectors, axial vectors, and second-order constitutive tensors are used to derive the transformed curl--curl equation and PEC trace directly from the original boundary-value problem. Under unique solvability, the common symmetry group of the problem data is inherited by the electric field. When the symmetry group of the electromagnetic medium admits a mirror symmetry, an arbitrary impressed current can be decomposed by the associated reflection projectors into mirror-even and mirror-odd components. The corresponding full-domain fields inherit the respective parities, and their exact restriction to one half-domain yields a perfect magnetic conductor (PMC) condition for the even field and a PEC condition for the odd field. The original problem is therefore represented exactly by two independent reduced-domain problems. A symmetry-compatible first-family N\'ed\'elec discretization preserves this decomposition at the discrete level. Analytical and numerical results verify the symmetry-inheritance theory, the exact PEC/PMC reduced-domain formulations, and the discrete reconstruction for constant and spatially varying anisotropic conductive media under both symmetric and asymmetric excitations.