  We study a simple nonlinear model defined on the honeycomb and triangular
lattices. We propose a bilinearization scheme for the field equations and
demonstrate that the resulting system is closely related to the well-studied
integrable models, such as the Hirota bilinear difference equation and the
Ablowitz-Ladik system. This result is used to derive the two sets of explicit
solutions: the N-soliton solutions and ones constructed of the Toeplitz
determinants.
