Abstract. We consider Hermite-Padé approximants in the framework of discrete integrable systems defined on the lattice Z 2 . We show that the concept of multiple orthogonality is intimately related to the Lax representations for the entries of the nearest neighbor recurrence relations and it thus gives rise to a discrete integrable system. We show that the converse statement is also true. More precisely, given the discrete integrable system in question there exists a perfect system of two functions, i.e., a system for which the entire table of Hermite-Padé approximants exists. In addition, we give a few algorithms to find solutions of the discrete system.