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Transformations $RS_4^2(3)$ of the ranks $\le 4$ and algebraic solutions of the sixth Painlevé equation. (English) Zbl 1019.34086
Considerable attention has been paid to the search of algebraic solutions to the sixth Painlevé equation. Here, compositions of rational transformations of independent variables of linear matrix ODEs with the Schlesinger transformations (RS-transformations) are used to construct algebraic solutions to Painlevé VI. RS-transformations of ranks $3$ and $4$ of $2\times 2$-matrix Fuchsian ODEs with $3$ singular points into analogous ODEs with $4$ singular points are classified.

34M55Painlevé and other special equations; classification, hierarchies
33E17Painlevé-type functions
34M25Formal solutions, transform techniques (ODE in the complex domain)
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