A symmetry reduction of the Painlevé IV hierarchy to the Flaschka-Newell Painlevé II hierarchy

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 Painlevé IV hierarchy. We show that the symmetry $Ψ(-λ)= σ_1 Ψ(λ) σ_1$ defines an invariant submanifold whose induced isomonodromic dynamics coincides with the Flaschka-Newell Painlevé 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 Painlevé 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.