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

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.