  A planar array of identical charges at vanishing temperature forms a Wigner
crystal with hexagonal symmetry. We take off one (reference) charge in a
perpendicular direction, hold it fixed, and search for the ground state of the
whole system. The planar projection of the reference charge should then evolve
from a six-fold coordination (center of a hexagon) for small distances to a
three-fold arrangement (center of a triangle), at large distances $d$ from the
plane. The aim of this paper is to describe the corresponding non-trivial
lattice transformation. For that purpose, two numerical methods (direct energy
minimization and Monte Carlo simulations), together with an analytical
treatment, are presented. Our results indicate that the $d=0$ and $d\to\infty$
limiting cases extend for finite values of $d$ from the respective starting
points into two sequences of stable states, with intersecting energies at some
value $d_t$; beyond this value the branches continue as metastable states.
