  We introduce a class of ${\mathbb{Z}}_N$ graded discrete Lax pairs, with
$N\times N$ matrices, linear in the spectral parameter. We give a
classification scheme for such Lax pairs and the associated discrete integrable
systems. We present two potential forms and completely classify the generic
case. Many well known examples belong to our scheme for $N=2$, so many of our
systems may be regarded as generalisations of these. Even at $N=3$, several new
integrable systems arise. Many of our equations are mutually compatible, so can
be used together to form "coloured" lattices.
  We also present continuous isospectral deformations of our Lax pairs, giving
compatible differential-difference systems, which play the role of continuous
symmetries of our discrete systems. We present master symmetries and a
recursive formulae for their respective hierarchies, for the generic case.
  We present two nonlocal symmetries of our discrete systems, which have a
natural representation in terms of the potential forms. These give rise to the
two-dimensional Toda lattice, with our nonlinear symmetries being the
B\"acklund transformations and our discrete system being the nonlinear
superposition formula (for the generic case).
