  This paper is concerned with the entire solutions of a two-dimensional
nonlocal periodic lattice dynamical system. With bistable assumption, it is
well known that the system has three different types of traveling fronts. The
existence of merging-front entire solutions originating from two fronts for the
system have been established by Dong, Li \& Zhang [{\it Comm. Pur Appl. Anal.},
{\bf17}(2018), 2517-2545]. Under certain conditions on the wave speeds, and by
some auxiliary rational functions with certain properties to construct
appropriate super- and sub solutions of the system, we establish two new types
of entire solutions which originating from three fronts.
