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 $E_{x} = (az +q(x))E$, where $a$ is a constant diagonal matrix, and $q(x)$ is an off-diagonal meromorphic function, for each $z \in \mathbb {C}$. 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 $q(x)$.