Takasaki's rational fourth Painlev\'e-Calogero system and geometric regularisability of algebro-Painlev\'e equations

TL;DR

This paper explores a Hamiltonian system related to the fourth Painlevé equation, demonstrating its geometric regularisation on rational surfaces and establishing connections with algebraic transformations and symmetries.

Contribution

It introduces a regularisation framework for a Painlevé-Calogero system lacking the Painlevé property, generalising Okamoto's spaces and relating the system to the fourth Painlevé equation through geometric transformations.

Findings

The Takasaki system admits a global Hamiltonian structure on a rational surface bundle.

The associated surface is related to the fourth Painlevé surface via blowdowns and a branched double cover.

Bäcklund transformations are lifted to algebraic symmetries of the Takasaki system.

Abstract

We study a Hamiltonian system without the Painlev\'e property and show that it admits a kind of regularisation on a bundle of rational surfaces with certain divisors removed, generalising Okamoto's spaces of initial conditions for the Painlev\'e differential equations. The system in question was obtained by Takasaki as part of the Painlev\'e-Calogero correspondence and possesses the algebro-Painlev\'e property, being related by an algebraic transformation to the fourth Painlev\'e equation. We provide an atlas for the bundle of surfaces in which the system has a global Hamiltonian structure, with all Hamiltonian functions being polynomial in coordinates just as in the case of Okamoto's spaces. We compare the surface associated with the Takasaki system with that of the fourth Painlev\'e equation, showing that they are related by a combination of blowdowns and a branched double cover, under which we lift the birational B\"acklund transformation symmetries of the fourth Painlev\'e equation to algebraic ones of the Takasaki system, including a discrete Painlev\'e equation. We also discuss and provide more examples in support of the idea that there is a connection between the algebro-Painlev\'e property and similar notions of regularisability, in an analogous way to how regular initial value problems for the Painlev\'e equations everywhere on Okamoto's spaces are related to the Painlev\'e property.