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Compositions of rational transformations of independent variables of linearmatrix ordinary differential equations (ODEs) with the Schlesingertransformations ($RS$-transformations) are used to construct algebraicsolutions of the sixth Painlev\'e equation. $RS$-Transformations of the ranks 3and 4 of $2\times2$ matrix Fuchsian ODEs with 3 singular points into analogousODE with 4 singular points are classified.Comment: 26 page

Transformations RS 4 2 (3) of the Ranks ≤ 4¶and Algebraic Solutions of the Sixth Painlevé Equation