On Initial Data in the Problem of Consistency on Cubic Lattices for $3\times 3$ Determinants

The paper is devoted to complete proofs of theorems on consistency on cubic lattices for $3\times 3$ determinants. The discrete nonlinear equations on ${\mathbb{Z}}^2$ defined by the condition that the determinants of all $3 \times 3$matricesofvaluesofthescalarfieldatthepointsofthelattice$$\mathbb{Z}$^2$thatformelementary$3 \times 3$ squares vanish are considered; some explicit concrete conditions of general position on initial data are formulated; and for arbitrary initial data satisfying these concrete conditions of general position, theorems on consistency on cubic lattices (a consistency “around a cube”) for the considered discrete nonlinear equations on ${\mathbb{Z}}^2$ defined by $3\times 3$ determinants are proved.