Hamiltonian Representation of Isomonodromic Deformations of Twisted Rational Connections: The Painlevé 1 Hierarchy

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 Painlevé 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 Painlevé 1 case \((g=1)\) and the next two elements of the Painlevé 1 hierarchy.