On the poles of Picard potentials

Abstract

We study the existence of a global meromorphic fundamental system of solutions for a system of two differential equations Ex = (az + q(x))E, where a is a constant diagonal matrix, and q(x) is an off-diagonal meromorphic function, for each z ∈ C. Following Gesztesy and Weikard (1998), who investigated this property of functions q(x) and its connection to finite-gap solutions of soliton equations, we call such q(x) Picard potentials. We obtain conditions for the Picard property of various potentials q(x).