On reducible monodromy representations of some generalized Lam\'e equation

TL;DR

This paper explicitly computes the monodromy data for a generalized Lamé equation with reducible but not completely reducible monodromy and solves the associated Riemann-Hilbert problem.

Contribution

It provides an explicit formula for the monodromy data in a specific reducible case and addresses the Riemann-Hilbert problem for this class of equations.

Findings

Explicit monodromy data formula derived

Riemann-Hilbert problem solved for the case

Enhanced understanding of reducible monodromy representations

Abstract

In this note, we compute the explicit formula of the monodromy data for a generalized Lam\'{e} equation when its monodromy is reducible but not completely reducible. We also solve the corresponding Riemman-Hilbert problem.