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arXiv · 2607.20106

Explicit Hamiltonian structure of the Flaschka-Newell Painlev\'{e} II hierarchy via symmetry reduction of the Painlev\'{e} IV hierarchy

Olivier Marchal·Mohamad Alameddine·Sergej Laub
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Abstract

We study the Hamiltonian structure of the Flaschka-Newell Painlev\'{e} II hierarchy via symmetry reduction of the associated space of meromorphic connections. Building on the realization of this hierarchy as a reduction of the Painlev\'{e} IV hierarchy via a $\mathbb{Z}_2$-symmetry, we construct a set of Darboux coordinates adapted to the involution. After a suitable change of trivialization, the symmetry acts diagonally in these coordinates, allowing the fixed-point locus to be explicitly described as a symplectic submanifold. This enables us to derive the reduced Hamiltonians after symmetry, thereby obtaining explicit expressions for the Hamiltonians and the Lax matrices of the Flaschka-Newell Painlev\'{e} II hierarchy. This strategy also illustrates how symmetry-adapted canonical Darboux coordinates enable explicit reductions of isomonodromic systems at the level of their underlying symplectic geometry.

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Olivier Marchal, Mohamad Alameddine, Sergej Laub. 2026-07-22. Explicit Hamiltonian structure of the Flaschka-Newell Painlev\'{e} II hierarchy via symmetry reduction of the Painlev\'{e} IV hierarchy. https://arxiv.org/abs/2607.20106

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