Geometric approach for the identification of Hamiltonian systems of quasi-Painlevé type

Marta Dell'Atti, Thomas Kecker

Published 2024-02-29, updated 2025-01-08Version 2

Some new Hamiltonian systems of quasi-Painlev\'e type are presented and the analogue of Okamoto's space of initial conditions computed. Using the geometric approach that was introduced originally for the identification problem of Painlev\'e equations, comparing the irreducible components of the inaccessible divisors arising in the blow-up process, we find bi-rational coordinate changes between some of these systems that give rise to the same global Hamiltonian structure. This scheme thus gives a method for identifying Hamiltonian systems up to bi-rational maps, which is performed in this article for systems of quasi-Painlev\'e type having singularities that are either square-root type algebraic poles or ordinary poles.