  In this paper we describe an intrinsically geometric way of producing
magnetic fields on $\S^3$ and $\R^3$ for which the corresponding Dirac
operators have a non-trivial kernel. In many cases we are able to compute the
dimension of the kernel. In particular we can give examples where the kernel
has any given dimension. This generalizes the examples of Loss and Yau (Commun.
Math. Phys. 104 (1986) 283-290).
