Integrable two-component systems of difference equations

We present two lists of two-component systems of integrable difference equations defined on the edges of the Z2 graph. The integrability of these systems is manifested by their Lax formulation which is a consequence of the multi-dimensional compatibility of these systems. Imposing constraints consistent with the systems of difference equations, we recover known integrable quad-equations including the discrete version of the Krichever-Novikov equation. The systems of difference equations give us in turn quadrirational Yang-Baxter maps.