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Integrable discrete scalar equations defined on a~two or a three dimensionallattice can be rewritten as difference systems in bond variables or in facevariables respectively. Both the difference systems in bond variables and thedifference systems in face variables can be regarded as vector versions of theoriginal equations. As a result, we link some of the discrete equations bydifference substitutions and reveal the non-potential versions of someconsistent-around-the-cube equations. We obtain higher-point configurations,including pairs of compatible six~points equations on the ${\mathbb Z}^2$lattice together with associated seven points equations. Also we obtain avariety of compatible ten point equations together with associated ten andtwelve point equations on the ${\mathbb Z}^3$ lattice. Finally, we presentintegrable multiquadratic quad relations.

Difference systems in bond and face variables and non-potential versions of discrete integrable systems