### Abstract

A Schroedinger operator with meromorphic potential is called monodromy-free if all solutions of the corresponding Schroedinger equation are meromorphic for all values of energy (so the corresponding monodromy in the complex plane is trivial). A nice class of examples is given by the so-called "finite-gap" operators, but in general the description of all monodromy-free operators is open even in the class of rational potentials, although in some special cases the answer is known (Duistermaat-Grunbaum, Gesztesy-Weikard, Oblomkov).

In the talk I will describe a class of Schroedinger operators with trivial monodromy, constructed in terms of the Painleve-IV transcendents and their higher analogues determined by the periodic dressing chains. We will discuss also a new interpretation and a fundamental role of the Stieltjes relations in this problem.