Note on the Singularity Reduction of Isomonodromy Systems Associated with Garnier Systems
Kohei Iwaki, Seiya Kato, Shotaro Sakurai
TL;DR
The work analyzes singularity reduction for isomonodromy systems tied to Garnier types Gar_{9/2} and Gar_{5/2+3/2}, proving that both admit a reduction to Schrödinger-type equations that remain isomonodromic with respect to the second Garnier time. It constructs two fourth-order ODEs describing the isomonodromic deformations; one matches a known DK14 equation and the other is a new example that lacks the Painlevé property, instead exhibiting Puiseux-type (quasi-Painlevé) singularities. In both cases, the classical limits yield genus-2 hyperelliptic curves, reinforcing a genus-2 geometry in the underlying spectral curve. The paper also clarifies how the Schrödinger form naturally implements suitable gauge choices, connecting isomonodromy deformation to the geometry of the reduced equations and highlighting gaps between isomonodromy and Painlevé properties. Overall, it extends the DK14 framework to Gar_{5/2+3/2}, supporting the conjecture that singularity reduction exists broadly for isomonodromy systems and pointing to a geometric interpretation via ramified irregular connections.
Abstract
In this paper, we study the isomonodromy systems associated with the Garnier systems of type 9/2 and type 5/2+3/2. We show that the both of isomonodromy systems admit the singularity reduction (restriction to a movable pole), and the resulting linear differential equations are isomonodromic with respect to the second variable of the Garnier systems. Furthermore, we find two fourth-order nonlinear ordinary differential equations that describe the isomonodromy deformation but lack the Painlevé property. One of these equations has been already found by Dubrovin--Kapaev in 2014, while the other provides a new example.