Uniqueness and non-uniqueness in the non-axisymmetric direction problem

We consider the following nonlinear boundary-value problem in the exterior space {${\hat{V}}={{\mathrm {x}} \ {\epsilon\mathbb{R}^{3}}:{\left | {\mathrm {x}} \right |} >1}$} of the unit sphere S: given a vector field ${S} \to{\mathbb{R}}$, we ask for all harmonic vector fields ${\hat{V}} \to{\mathbb{R}}$ vanishing at infinity and parallel to D on S, that is, there is an amplitude $a:{S} \to{\mathb...