How to generate all possible rational Wilf-Zeilberger pairs?

A Wilf–Zeilberger pair $(F,G)$ in the discrete case satisfies the equation $F(n+1,k)-F(n,k)=G(n,k+1)-G(n,k)$. We present a structural description of all possible rational Wilf–Zeilberger pairs and their continuous and mixed analogues.