  We consider a class of systems of difference equations defined on an
elementary quadrilateral of the ${\mathbb{Z}}^2$ lattice, define their
eliminable and dynamical variables, and demonstrate their use. Using the
existence of infinite hierarchies of symmetries as integrability criterion, we
derive necessary integrability conditions and employ them in the construction
of the lowest order symmetries of a given system. These considerations are
demonstrated with the help of three systems from the class of systems under
consideration.
