  We carry out the generalized symmetry classification of polylinear autonomous
discrete equations defined on the square, which belong to a twelve-parametric
class. The direct result of this classification is a list of equations
containing no new examples. However, as an indirect result of this work we find
a number of integrable examples pretending to be new. One of them has a
nonstandard symmetry structure, the others are analogues of the Liouville
equation in the sense that those are Darboux integrable. We also enumerate all
equations of the class, which are linearizable via a two-point first integral,
and specify the nature of integrability of some known equations.
