On some examples of identifying Painlev\'e equations using the geometry of the Okamoto space of initial conditions

TL;DR

This paper demonstrates how algebro-geometric methods can be used to identify specific Painlevé equations by explicitly transforming and matching them with standard forms and Hamiltonians.

Contribution

It provides explicit coordinate transformations and Hamiltonian identifications for new Painlevé equations derived from isomonodromic deformation studies.

Findings

Explicit transformations linking new equations to standard Painlevé forms

Identification of Hamiltonians for the considered equations

Application of algebro-geometric theory to Painlevé classification

Abstract

In this short note we give two examples of using the algebro-geometric theory of Painlev\'e equations to solve the Painlev\'e identification problem. The equations that we consider were recently obtained by M. van der Put and J. Top in their study of a certain ansatz of isomonodromic deformations of linear ODEs. We provide explicit coordinate transformations identifying these examples with standard form of some Painlev\'e equations and also explicitly identify their Hamiltonians.