Schlesinger transformations for algebraic Painlevé VI solutions

Schlesinger (S) transformations can be combined with a direct rational (R) pull-back of a hypergeometric $2\times 2$ system of ODEs to obtain $RS^2_4$-pullback transformations to isomonodromic $2\times 2$ Fuchsian systems with 4 singularities. The corresponding Painlevé VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. This article demonstrates direct computations (involving polynomial syzygies) of Schlesinger transformations that affect several singular points at once, and presents an algebraic procedure of computing algebraic Painlevé VI solutions without deriving full RS-pullback transformations.