Some properties of the potential theory operators and and their application to investigation of the basic electro- and magnetostatic equation

Some new properties of the double layer potential direct value on $S=\partial \Omega$ operator $B^*$ are proved. In particular the existence in $H^{1/2}(S)$ of a basis, consisting of $B^*$ eigen functions, is shown. Basing on these properties an equivalence of the vector integral equation
$$ \alpha \mathbf M(x)+\nabla \int _\Omega \mathbf M(y)\nabla _y|x-y|\,dy=\mathbf H(x), \qquad \alpha \geqslant 0,\quad \Omega \subset R^3,$$
to the known scalar equation with the operator $B^*$ is proved. This vector equation arisis in the integral formulation of the electro- and magnetostatic field problem. The properties of the left-hand side operator and solutions of the equation are investigated.