# Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlev\'{e} $1$ hierarchy

## TL;DR

This paper develops a Hamiltonian framework and explicit Lax pairs for the Painlevé 1 hierarchy using twisted rational connections, providing new formulas and coordinate transformations for analyzing isomonodromic deformations.

## Contribution

It introduces explicit Hamiltonians and Lax pairs for the Painlevé 1 hierarchy via twisted connections, and constructs coordinate changes to simplify their polynomial structure.

## Key findings

- Explicit formulas for Lax pairs and Hamiltonians in terms of irregular times.
- A reduction map for the space of irregular times to fewer deformations.
- Symplectic coordinate transformations yielding polynomial Hamiltonians.

## Abstract

In this paper, we build the Hamiltonian system and the corresponding Lax pairs associated to a twisted connection in $\mathfrak{gl}_2(\mathbb{C})$ admitting an irregular and ramified pole at infinity of arbitrary degree, hence corresponding to the Painlev\'{e} $1$ hierarchy. We provide explicit formulas for these Lax pairs and Hamiltonians in terms of the irregular times and standard $2g$ Darboux coordinates associated to the twisted connection. Furthermore, we obtain a map that reduces the space of irregular times to only $g$ non-trivial isomonodromic deformations. In addition, we perform a symplectic change of Darboux coordinates to obtain a set of symmetric Darboux coordinates in which Hamiltonians and Lax pairs are polynomial. Finally, we apply our general theory to the first cases of the hierarchy: the Airy case $(g=0)$, the Painlev\'{e} $1$ case $(g=1)$ and the next two elements of the Painlev\'{e} $1$ hierarchy.