Monondromy-free matrix Zakharov–Shabat operators

A pair $u,v$ of meromorphic matrix-valued $n\times n$-germs at a point $x_0$ is said to determine a monondromy-free matrix Zakharov–Shabat operator at this point if, for every complex $z$, all solutions $f,g$ of the linear system of ordinary differential equations $f'(x)=zf(x)+u(x)g(x), g'(x)=v(x)f(x)-zg(x)$ are meromorphic at this point. Which conditions on the Laurent coefficients of $u,v$ at $x_0$ are necessary and/or sufficient for this property to hold? We survey known results on this theme and their applications to global meromorphic extension of solutions of matrix soliton equations and reduce the proof of a criterion for monodromy-free operators (in the case when $n=1$ and $u,v$ have simple poles at $x_0$) to an interesting technical assertion about sequences of real numbers.