Exact Solution for an Insulating Circular Hole in a Two-Dimensional Anisotropic Medium

We present the exact analytic solution for the problem of an insulating circular hole immersed in an infinite two-dimensional anisotropic conducting medium, using the coordinate-free geometric-distributional formalism developed in Article I. The solution strategy employs an affine transformation (rescaling) that maps the anisotropic Laplace equation into the standard isotropic Laplace equation, deforming the circular hole into a virtual ellipse whose eccentricity directly encodes the medium anisotropy. Maxwell's classical solution for the insulating ellipse in elliptic coordinates is then transformed back to physical space, yielding a closed-form expression for the induced surface charge density on the hole boundary. The analysis reveals: (i) the existence of exactly two neutral nodes (points of zero charge), whose angular positions migrate continuously with the anisotropy ratio; (ii) in the extreme hyperbolic limit (transverse conductivity approaching zero), the charge density exhibits local non-integrable divergences at the frontal collision points, characterizing a divergent ideal capacitor scenario; and (iii) in this limit, the neutral node positions become independent of the detailed obstacle geometry, constituting a topological invariant of the hyperbolic regime. Together with Article I, this work provides a comprehensive framework for anisotropic transport in media with singular boundaries, with applications to modern two-dimensional materials (black phosphorus, transition metal dichalcogenides) and hyperbolic metamaterials