Investigation of an equation of electrophysics

The vector integral equation
$$ \alpha\mathbf M+\nabla\int_\Omega\mathbf M\nabla|x-y|^{-1}\,dy=\mathbf H,\qquad\alpha\ge0,\quad\Omega\subset\mathbb R^3, $$
encountered in classical problems of electrostatics and magnetostatics, is investigated. The structure and properties of its solutions are investigated by studying the corresponding integral operator, and a numerical method is proposed.