Discretising systematically the Painlevé equations

We apply a method that we have already proposed for the systematic discretisation of differential systems to the construction of discrete analogues of the Painlevé equations. The important point is that in the case of the Painlevé equations one needs to preserve the integrability. We show that this is indeed possible using our systematic procedure applied to the equations obtained from the Hamiltonian description of the Painlevé equations.