Cubic and quartic transformations of the sixth Painleve equation in terms of Riemann-Hilbert correspondence

TL;DR

This paper classifies quadratic polynomial transformations of the monodromy manifold for Painleve VI, identifying known transformations and discovering a new degree 3 transformation of Picard's solutions.

Contribution

It provides a classification of quadratic transformations of the monodromy manifold and introduces a novel degree 3 transformation of Painleve VI solutions.

Findings

Identified three quadratic polynomial transformations of the monodromy manifold.

Two transformations correspond to known quadratic or quartic Painleve VI transformations.

Discovered a new degree 3 transformation of Picard's solutions.

Abstract

A starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2x2 isomonodromic Fuchsian systems associated to the Painleve VI equation. Up to birational automorphisms of the monodromy manifold, we find three transformations. Two of them are identified as the action of known quadratic or quartic transformations of the Painleve VI equation. The third transformation of the monodromy manifold gives a new transformation of degree 3 of Picard's solutions of Painleve VI.