  We generalize the study of Heisenberg uniqueness pairs considered in earlier
work with Alfonso Montes-Rodriguez. We also find the critical density in the
case of one branch of the hyperbola. In the critical case there is
non-uniqueness for a one-dimensional space of solutions. These solutions cannot
vanish at any other point of the axes of the characteristic lattice-cross, due
to an unexpected connection with the Nielsen spiral.
