We discretise the Lax pair for reduced Nahm systems and prove its equivalence with Kahan–Hirota–Kimura discretisation procedure. show that these pairs guarantee integrability of discrete providing an invariant. Also, we cannot solve general problem characterisation discretisations.

We obtain 3 × 3 matrix Lax pairs for systems of ODEs that are solvable in terms of the fourth, fifth and sixth Painlevé equations by considering similarity reductions of the scattering Lax pair for the (2+1)-dimensional three-wave resonant interaction system. These results allow us to construct new 3× 3 Lax representations for the fourth and fifth Painlevé equations, together with the previousl...

In this article, we prove that we can introduce a small ~ parameter in the six Painlevé equations through their corresponding Lax pairs and Hamiltonian formulations. Moreover, we prove that these ~-deformed Lax pairs satisfy the Topological Type property proposed by Bergère, Borot and Eynard for any generic choice of the monodromy parameters. Consequently we show that one can reconstruct the fo...