A discrete iso-spectral problem with positive and negative hierarchies and integrable coupling system

Two hierarchies of integrable positive and negative nonlinear lattice systems are derived from a discrete iso-spectral problem. When the Lax operators are expanded by virtue of the positive and negative power expansion with respect to the spectral parameter, we get the corresponding integrable hierarchies. Moreover, a direct matrix spectral method is used to get the associated integrable coupling system of the first resulting hierarchy.