In this article we present the results obtained applying the multiple scale
expansion up to the order \epsilon^6 to a dispersive multilinear class of
equations on a square lattice depending on 13 parameters. We show that the
integrability conditions given by the multiple scale expansion give rise to 4
nonlinear equations, 3 of which are new, depending at most on 2 parameters and
containing integrable sub cases. Moreover at least one sub case provides an
example of a new integrable system
