Symmetries of differential-difference dynamical systems in a two-dimensional lattice

Classification of differential-difference equation of the form ${\ddot{u}}_{nm}=F_{nm}(t,\{u_{pq}\}{\mathrm{\mid }}_{(p,q)\in \mathrm{\Gamma }})$ are considered according to their Lie point symmetry groups. The set $\mathrm{\Gamma }$ represents the point $(n,m)$ and its six nearest neighbors in a two-dimensional triangular lattice. It is shown that the symmetry group can be at most 12-dimensional for abelian symmetry algebras and 13-dimensional for nonsolvable symmetry algebras.