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arXiv · 2606.24662

A symmetry reduction of the Painlev\'{e} IV hierarchy to the Flaschka-Newell Painlev\'{e} II hierarchy

Mohamad Alameddine·Olivier Marchal
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Abstract

We study the isomonodromic deformation problem associated with rank-two meromorphic connections on the Riemann sphere having one regular singularity and one irregular singularity of even order at infinity, corresponding to the even Painlev\'{e} IV hierarchy. We show that the symmetry $\Psi(-\lambda)= \sigma_1 \Psi(\lambda) \sigma_1$ defines an invariant submanifold whose induced isomonodromic dynamics coincides with the Flaschka-Newell Painlev\'{e} II hierarchy. Under this identification, the corresponding Lax matrices, Darboux coordinates and Hamiltonian structures can be matched explicitly. In particular, the Hamiltonians of the first members of the Flaschka-Newell hierarchy are recovered from the even Painlev\'{e} IV hierarchy. This provides a geometric interpretation of the Flaschka-Newell hierarchy as a symmetry reduction of an isomonodromic deformation problem, complementing its classical description as a similarity reduction of the modified Korteweg-de Vries hierarchy.

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Mohamad Alameddine, Olivier Marchal. 2026-06-23. A symmetry reduction of the Painlev\'{e} IV hierarchy to the Flaschka-Newell Painlev\'{e} II hierarchy. https://arxiv.org/abs/2606.24662

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