  A general elliptic $N\times N$ matrix Lax scheme is presented, leading to two
classes of elliptic lattice systems, one which we interpret as the higher-rank
analogue of the Landau-Lifschitz equations, while the other class we
characterize as the higher-rank analogue of the lattice Krichever-Novikov
equation (or Adler's lattice). We present the general scheme, but focus mainly
of the latter type of models. In the case $N=2$ we obtain a novel Lax
representation of Adler's elliptic lattice equation in its so-called 3-leg
form. The case of rank $N=3$ is analysed using Cayley's hyperdeterminant of
format $2\times2\times2$, yielding a multi-component system of coupled 3-leg
quad-equations.
