The geometry of Chazy's homogeneous third-order differential equations

TL;DR

This paper classifies the geometric structures associated with Chazy's homogeneous third-order differential equations, revealing their connection to affine surfaces with single-valued solutions, and relates these findings to recent mathematical results.

Contribution

It provides a classification of the affine surfaces arising from Chazy's equations and links these geometric structures to existing research on holomorphic vector fields.

Findings

Classification of affine surfaces associated with Chazy's equations

Connection between these surfaces and single-valued solutions

Relation to recent work by Rebelo and the author

Abstract

Chazy studied a family of homogeneous third-order autonomous differential equations. They are those, within a certain class, admitting exclusively single-valued solutions. Each one of these equations yields a polynomial vector field in complex three-dimensional space. For almost all of these these vector fields, the Zariski closure of a generic orbit yields an affine surface endowed with a holomorphic vector field that has exclusively single-valued solutions. We classify these surfaces and relate this classification to recent results of Rebelo and the author.