The Painlevé divisors of the autonomous 4-dimensional Painlevé-type equations (Mathematical structures of integrable systems and their applications)

Abstract

Following [1], we consider compactification of the Liouville tori for the autonomous 4-dimensional Painlevé-type equations by adjoining the Painlevé divisors. We can see that thegenus 2 components of the Painlevé divisors are all isomorphic to the corresponding spectralcurve. We illustrate the motivating examples for [21], which asserts that the genus 2 curvesare uniquely determined by the Liouville tori of the autonomous 4-dimensional Painlevé-typeequations. This may enable us to recover a linear problem from the nonlinear integrablesystems.