  At large values of the anisotropy \Delta, the open-boundary Heisenberg
spin-1/2 chain has eigenstates displaying localization at the edges. We present
a Bethe ansatz description of this `edge-locking' phenomenon in the entire
\Delta>1 region. We focus on the simplest spin sectors, namely the highly
polarized sectors with only one or two overturned spins, i.e., one-particle and
two-particle sectors. Edge-locking is associated with pure imaginary solutions
of the Bethe equations, which are not commonly encountered in periodic chains.
In the one-particle case, at large anisotropies there are two eigenstates with
imaginary Bethe momenta, related to localization at the two edges. For any
finite chain size, one of the two solutions become real as the anisotropy is
lowered below a certain value. For two particles, a richer scenario is
observed, with eigenstates having the possibility of both particles locked on
the same or different edge, one locked and the other free, and both free either
as single magnons or as bound composites corresponding to `string' solutions.
For finite chains, some of the edge-locked spins get delocalized at certain
values of the anisotropy (`exceptional points'), corresponding to imaginary
solutions becoming real. We characterize these phenomena thoroughly by
providing analytic expansions of the Bethe momenta for large chains, large
anisotropy, and near the exceptional points. In the large-chain limit all the
exceptional points coalesce at the isotropic point (\Delta=1) and edge-locking
becomes stable in the whole \Delta>1 region.
