Algebraic solutions of the sixth Painleve equation

TL;DR

This paper classifies all algebraic solutions of the Painleve VI equation by analyzing the finite orbits of a modular group action on SL(2,C)-triples, providing a comprehensive algebraic characterization.

Contribution

It introduces a method to classify algebraic solutions of Painleve VI using group action orbit analysis, a novel approach in the field.

Findings

Complete classification of algebraic solutions of Painleve VI.

Identification of finite orbits of the modular group action.

Connection between group orbits and algebraic solutions.

Abstract

We describe all finite orbits of an action of the extended modular group $\bar{\Lambda}$ on conjugacy classes of SL(2,C)-triples. The result is used to classify all algebraic solutions of the general Painleve VI equation up to parameter equivalence.