Singularities or emergent losses in radially uniaxial spheres

To find the electrostatic solution of a radially uniaxially anisotropic sphere in a constant external field is fairly straightforward. However, the resulting quasistatic polarizability exhibits many interesting properties including singularities and dubious emergent losses. By removing the origin and adding small losses, it turns out that the solution is valid in a certain limiting sense. Without puncturing the sphere, the potential is singular at the origin if the permittivity is indefinite, and for certain ranges of parameter values infinitesimally small losses give rise to significant effective losses.