Boussinesq-like multi-component lattice equations and multi-dimensional consistency

Various classes of one-component lattice equations, defined by a multilinear relation between values at the vertices of an elementary square, have recently been classified using the requirement of multi-dimensional consistency (consistency-around-the-cube, CAC). Here we consider multi-component equations, with some equations defined on the edges of the consistency cube and others on the faces of the cube. Some examples of this type are already known, including the lattice-modified Boussinesq equation (lmBSQ). We classify the edge equations into three canonical forms and derive the consequences of their CAC-property. This restricts the form of the face equation sufficiently so that its CAC-property can be analyzed. As a result we obtain a number of integrable multi-component lattice equations, some generalizing lmBSQ.