Note on the Singularity Reduction of Isomonodromy Systems Associated with Garnier Systems

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 Painlev\'e property. One of these equations has been already found by Dubrovin--Kapaev in 2014, while the other provides a new example.