  Potentials play an important role in solving boundary value problems for
elliptic equations. In the middle of the last century, a potential theory was
constructed for a two-dimensional elliptic equation with one singular
coefficient. In the study of potentials, the properties of the fundamental
solutions of the given equation are essentially and fruitfully used. At the
present time, fundamental solutions of a three-dimensional elliptic equation
with one degeneration line are already known. In this paper, we investigate the
double- and simple-layer potentials for this kind of elliptic equations.
Results from potential theory allow us to represent the solution of the
boundary value problems in integral equation form. By using some properties of
Gaussian hypergeometric function, we prove limiting theorems and derive
integral equations concerning a densities of the double- and simple-layer
potentials. The obtained results are applied to find an explicit solution of
the Dirichlet and Holmgren problems for the three-dimensional singular elliptic
equation in the half of the ball.
