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Solving the electroencephalography (EEG) forward problem is a fundamental step in a wide range of applications including biomedical imaging techniques based on inverse source localization. State-of-the-art electromagnetic solvers resort to a computationally expensive volumetric discretization of the full head to account for its complex and heterogeneous electric profile. The more efficient, popular in biomedical imaging circles, but unfortunately oversimplifying Boundary Element Method (BEM) relies instead on a piecewise-uniform approximation that severely curbs its application in high resolution EEGs. This contribution lifts the standard BEM contraints by treating the local anisotropies with adequate wire and thin volume integral equations that are tailored to specific structures of the fibrous white matter and the inhomogeneous skull. The proposed hybrid integral equation formulation thereby avoids the full volumetric discretization of the head medium and allows for a realistic and efficient BEM-like solution of the anisotropic EEG forward problem. The accuracy and flexibility of the proposed formulation is demonstrated through numerical experiments involving both canonical and realistic MRI-based head models.