Darboux integrable discrete equations possessing an autonomous first-order integral

Darboux integrable difference equations on the quad-graph are completely described in the case of the equations that possess autonomous first-order integrals in one of the characteristics. A generalization of the discrete Liouville equation is obtained from a subclass of this equation via a non-point transformation. The detailed proof of the general proposition on the symmetry structure for the quad-graph equations is given as an auxiliary result.