  We study solitons arising in a system describing the interaction of a
two-dimensional discrete hexagonal lattice with an additional electron field
(or, in general, an exciton field). We assume that this interaction is
electron-phonon-like. In our previous paper [4], we have studied the existence
of two-dimensional solitons and have found that these solitons exist only if
the electron-phonon coupling constant is sufficiently large. In this paper, we
report the results of our investigation for small values of this constant,
close to its critical value for the existence of solitons. We find that as the
coupling decreases the soliton gets very broad and then becomes effectively
one-dimensional.
