  Compositions of rational transformations of independent variables of linear
matrix ordinary differential equations (ODEs) with the Schlesinger
transformations ($RS$-transformations) are used to construct algebraic
solutions of the sixth Painlev\'e equation. $RS$-Transformations of the ranks 3
and 4 of $2\times2$ matrix Fuchsian ODEs with 3 singular points into analogous
ODE with 4 singular points are classified.
