  In this paper we provide a new method for establishing the rotational
symmetry of the solutions to a couple of very classical overdetermined problems
arising in potential theory, in both the exterior and the interior punctured
domain. Thanks to a conformal reformulation of the problems, we obtain
Riemannian manifolds with zero Weyl tensor satisfying a quasi-Einstein type
equation. Exploiting these geometric properties, we conclude via a splitting
argument that the manifolds obtained are half cylinders. In turn, the
rotational symmetry of the potential is implied. To the authors' knowledge,
some of the overdetermining conditions considered here are new.
