On the Monodromy-Preserving Deformation of a Double Confluent Heun Equation

We consider the second-order linear differential equation presented in Subsection 4.8 of the paper by J. Dereziński, A. Ishkhanyan, and A. Latosiński [SIGMA 17, 056 (2021)] and called there the deformed double confluent Heun equation. We prove that the additional singular point arising during the construction of the equation does not affect the analytic structure of its solution space. Moreover, we prove that under certain conditions a one-parameter transformation of the equation, called a deformation, with coefficients expressed in terms of the third Painlevé transcendent, leaves its monodromy unchanged. The proofs are self-contained.