On Stieltjes relations, Painlev\'e-IV hierarchy and complex monodromy

A generalisation of the Stieltjes relations for the Painlev\'e-IV transcendents and their higher analogues determined by the dressing chains is proposed. It is proven that if a rational function from a certain class satisfies these relations it must be a solution of some higher Painlev\'e-IV equation. The approach is based on the interpretation of the Stieltjes relations as local trivial monodromy conditions for certain Schr\"odinger equations in the complex domain. As a corollary a new class of the Schr\"odinger operators with trivial monodromy is constructed in terms of the Painlev\'e-IV transcendents.