  Removing al least one point from the unit sphere in $ R^{3}$ allows to
factorize the angular part of the laplacian with a Cauchy-Riemann type
operator. Solutions to this operator define a complex algebra of potential
functions. A family of these solutions is shown to be normalizable on the
sphere so it is possible to construct associate solutions for every radial
solution to the time-independant Schr\"odinger equation with a radial
potential, such that this family of solutions is square integrable in $R^{3}$.
While this family of associated solutions are singular on at least one
half-plane, they are square-integrable in almost all of $R^{3}$.
