  We consider overdetermined systems of difference equations for a single
function $u$ which are consistent, and propose a general framework for their
analysis. The integrability of such systems is defined as the existence of
higher order symmetries in both lattice directions and various examples are
presented. Two hierarchies of consistent systems are constructed, the first one
using lattice paths and the second one as a deformation of the former. These
hierarchies are integrable and their symmetries are related via Miura
transformations to the Bogoyavlensky and the discrete Sawada-Kotera lattices,
respectively.
