Leading terms of asymptotic expansions for the general complex solutions of the fifth Painlevé equation as t → ı∞ are found. These asymptotics are parameterized by monodromy data of the associated linear ODE,The parametrization allows one to derive connection formulas for the asymptotics. We provide numerical verification of the results. Important special cases of the connection formulas are also considered.

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 Painlevé equation. RS-Transformations of the ranks 3 and 4 of 2 × 2 matrix Fuchsian ODEs with 3 singular points into analogous ODE with 4 singular points are classified.

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A combination of rational mappings and Schlesinger transformations for a matrix form of the hypergeometric equation is used to construct higher order transformations for the Gauss hypergeometric function.

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