Consider the square lattice Z 2 with vertices at points with integer-valued coordinates in R 2 = {(x 1 , x 2 )| x k ∈ R, k = 1, 2} and complex (or real) scalar fields u on the lattice Z 2 , u : Z 2 → C, that are defined by their values u i 1 i 2 , u i 1 i 2 ∈ C, at each vertex of the lattice with the coordinates (i 1 , i 2 ), i k ∈ Z, k = 1, 2. Consider a class of two-dimensional discrete equations on the lattice Z 2 for the field u that are defined by functions Q(x 1 , x 2 , x 3 , x 4 ) of four variables with the help of the relations