Special polynomials related to the supersymmetric eight-vertex model. II. Schr\"odinger equation

We show that symmetric polynomials previously introduced by the author satisfy a certain differential equation. After a change of variables, it can be written as a non-stationary Schr\"odinger equation with elliptic potential, which is closely related to the Knizhnik--Zamolodchikov--Bernard equation and to the canonical quantization of the Painlev\'e VI equation. In a subsequent paper, this will be used to construct a four-dimensional lattice of tau functions for Painlev\'e VI.