In non-relativistic quantum mechanics, singular potentials in problems with spherical symmetry lead to a Schrödinger equation for stationary states with non-Fuchsian singularities both as and as . In the 1960s, an analytic approach was developed for the investigation of scattering from such potentials, with emphasis on the polydromy of the wavefunction in the r-variable. This paper extends those early results to an arbitrary number of spatial dimensions. The Hill-type equation which leads, in principle, to the evaluation of the polydromy parameter, is obtained from the Hill equation for a two-dimensional problem by means of a simple change of variables. The asymptotic forms of the wavefunction as and as are also derived. The Darboux technique of intertwining operators is then applied to obtain an algorithm that makes it possible to solve the Schrödinger equation with a singular potential admitting a Laurent expansion, if the exact solution with even just one term is already known.