  Reconstructing magnetizations from measurements of the generated magnetic
potential is generally non-unique. The non-uniqueness still remains if one
restricts the magnetization to those induced by an ambient magnetic dipole
field (i.e., the magnetization is described by a scalar susceptibility and the
dipole direction). Here, we investigate the situation under the additional
constraint that the susceptibility is either spatially localized in a subregion
of the sphere or that it is band-limited. If the dipole direction is known,
then the susceptibility is uniquely determined under the spatial localization
constraint while it is only determined up to a constant under the the
assumption of band-limitedness. If the dipole direction is not known,
uniqueness is lost again. However, we show that all dipole directions that
could possibly generate the measured magnetic potential need to be zeros of a
certain polynomial which can be computed from the given potential. We provide
examples of non-uniqueness of the dipole direction and examples on how to find
admissible candidates for the dipole direction under the spatial localization
constraint.
