  We consider quasilinear, multi-variable, constant coefficient, lattice
equations defined on the edges of the elementary square of the lattice, modeled
after the lattice modified Boussinesq (lmBSQ) equation, e.g., $\tilde y
z=\tilde x-x$. These equations are classified into three canonical forms and
the consequences of their multidimensional consistency
(Consistency-Around-the-Cube, CAC) are derived. One of the consequences is a
restriction on form of the equation for the $z$ variable, which in turn implies
further consistency conditions, that are solved. As result we obtain a number
of integrable multi-component lattice equations, some generalizing lmBSQ.
