An integrable multicomponent quad equation and its Lagrangian   formulation
  We present a hierarchy of discrete systems whose first members are the lattice modified Korteweg-de Vries equation, and the lattice modified Boussinesq equation. The N-th member in the hierarchy is an N-component system defined on an elementary plaquette in the 2-dimensional lattice. The system is multidimensionally consistent and a Lagrangian which respects this feature, i.e., which has the desirable closure property, is obtained. 
J. Atkinson, S. B. Lobb and F. W. Nijhoff
nlin.SI