Monodromy free Schrödinger operators and affine ${\mathfrak{s}\mathfrak{l}}_2$ master functions

Given a non-zero polynomial $P(x)$, we study Fuchsian differential operators of the form $L={\mathrm{\partial }}_x^2-u(x)$ such that for all $\lambda \in \mathbb{C}$ the operator $L+\lambda P(x)$ is monodromy free. We prove that all such operators are obtained from populations of critical points of ${\widehat{\mathfrak{s}\mathfrak{l}}}_2$ master functions. Moreover, we show that the reproduction procedure of critical points corresponds to a Darboux transformation of operator $P^{-1}(x)L$. As a result, we obtain a classification of all operators $L$ with such properties in the case of $P(x)=x^k$.