The $2{\times}2$ matrix Schlesinger system and the Belavin–Polyakov–Zamolodchikov system

We show that the Belavin–Polyakov–Zamolodchikov equation of the minimal model of conformal field theory with the central charge $c=1$ for the Virasoro algebra is contained in a system of linear equations that generates the Schlesinger system with $2{\times}2$ matrices. This generalizes Suleimanov's result on the Painlevé equations. We consider the properties of the solutions, which are expressible in terms of the Riemann theta function.