A new two--parameter family of isomonodromic deformations over the five punctured sphere

TL;DR

This paper constructs a two-parameter family of logarithmic flat connections with dihedral monodromy on the complex projective plane, leading to new algebraic solutions of Garnier systems and insights into related foliations.

Contribution

It introduces an explicit two-parameter family of isomonodromic deformations with dihedral monodromy, connecting flat connections, Garnier systems, and Lotka-Volterra foliations.

Findings

Derived algebraic solutions of Garnier systems.

Constructed a non-generic family of transversally projective Lotka-Volterra foliations.

Established explicit connections with dihedral monodromy on the quintic locus.

Abstract

The object of this paper is to describe an explicit two--parameter family of logarithmic flat connections over the complex projective plane. These connections have dihedral monodromy and their polar locus is a prescribed quintic composed of a circle and three tangent lines. By restricting them to generic lines we get an algebraic family of isomonodromic deformations of the five--punctured sphere. This yields new algebraic solutions of a Garnier system. Finally, we use the associated Riccati one--forms to construct an interesting non--generic family of transversally projective Lotka--Volterra foliations.