  We describe the closures of locally divergent orbitsunder the action of tori
on Hilbert modular spaces of rank r = 2. In particular, we prove that if D is a
maximal R-split torus acting on a real Hilbert modular space then every locally
divergent non-closed orbit is dense for r > 2 and its closure is a finite union
of tori orbits for r = 2. Our results confirm an orbit rigidity conjecture of
Margulis in all cases except for (i) r = 2 and, (ii) r > 2 and the Hilbert
modular space corresponds to a CM-field; in the cases (i) and (ii) our results
contradict the conjecture. As an application, we describe the set of values at
integral points of collections of non-proportional, split, binary, quadratic
forms over number fields.
