  Various Schlesinger transformations can be combined with a direct pull-back
of a hypergeometric 2x2 system to obtain $RS^2_4$-pullback transformations to
isomonodromic 2x2 Fuchsian systems with 4 singularities. The corresponding
Painleve VI solutions are algebraic functions, possibly in different orbits
under Okamoto transformations. This paper demonstrates a direct computation of
Schlesinger transformations acting on several apparent singular points, and
presents an algebraic procedure (via syzygies) of computing algebraic Painleve
VI solutions without deriving full RS-pullback transformations.
