Transformations ${RS}_4^2(3)$ of the Ranks $\le 4$ and Algebraic Solutions of the Sixth Painlevé Equation

Abstract

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 $2\times 2$ matrix Fuchsian ODEs with 3 singular points into analogous ODE with 4 singular points are classified.

26 pages

References in corpus (2)

• Special Functions of the Isomonodromy Type, Rational Transformations of Spectral Parameter, and Algebraic Solutions of the Sixth Painlevé Equation
• Non-Schlesinger Deformations of Ordinary Differential Equations with Rational Coefficients

Cited by in corpus (18)

• Some explicit solutions to the Riemann-Hilbert problem
• Computation of highly ramified coverings
• Six results on Painleve VI
• Dessins d'Enfants, Their Deformations and Algebraic the Sixth Painlevé and Gauss Hypergeometric Functions
• Geometry of moduli spaces of meromorphic connections on curves, Stokes data, wild nonabelian Hodge theory, hyperkahler manifolds, isomonodromic deformations, Painleve equations, and relations to Lie theory
• On a class of algebraic solutions to Painlevé VI equation, its determinant formula and coalescence cascade
• Classification of algebraic solutions of irregular Garnier systems
• On Algebraic Solutions to Painleve VI
• Canonical structure and symmetries of the Schlesinger equations
• Remarks Towards the Classification of $R{S}_4^2(3)$-Transformations and Algebraic Solutions of the Sixth Painlevé Equation
• Schlesinger transformations for algebraic Painleve VI solutions
• Special Solutions of the Sixth Painleve Equation with Solvable Monodromy
• Finite branch solutions to Painleve VI around a fixed singular point
• Solutions algébriques. Solutions algébriques partielles des équations isomonodromiques sur les courbes de genre $2$
• Computation of RS-pullback transformations for algebraic Painleve VI solutions
• Construction et classification de certaines solutions algébriques des systèmes de Garnier
• Differential relations for almost Belyi maps
• Algebraic isomonodromic deformations of logarithmic connections on the Riemann sphere and finite braid group orbits on character varieties