We study the special limits of discrete Painlevé equations belonging to the q-PVI and d-PV families, when the independent variable goes to 1 or 0, respectively. We obtain discrete systems which are shown to be either contiguities of solutions of continuous Painlevé equations, usually of PVI...

We examine critically the Gambier equation and show that it is the generic linearisable equation containing, as reductions, all the second-order equations which are integrable through linearisation. We then introduce the general discrete form of this equation, the Gambier mapping, and present...

The discovery of the Kowalevski top (1889) as a new integrable system posed the question whether there exists a rigorous relation between integrability of a Hamiltonian system and the analytic property of the solution in the complex time plane. Many examples suggest a hidden relation between the...