Global branching of solutions to ODEs and integrability

TL;DR

This paper generalizes the Painlevé property to classify integrable ODEs, including the Lane-Emden equation, by analyzing the global branching behavior of solutions and their Riemann surfaces.

Contribution

It introduces a generalized integrability criterion based on solution branching and classifies certain first- and second-order ODEs with this property.

Findings

Identified integrable cases of the Lane-Emden equation.

Classified first-order ODEs with finite-sheet solution Riemann surfaces.

Established necessary conditions for a broad class of second-order equations.

Abstract

We consider a natural generalisation of the Painlev\'e property and use it to identify the known integrable cases of the Lane-Emden equation with a real positive index. We classify certain first-order ordinary differential equations with this property and find necessary conditions for a large family of second-order equations. We consider ODEs such that, given any simply connected domain $\Omega$ not containing fixed singularities of the equation, the Riemann surface of any solution obtained by analytic continuation along curves in $\Omega$ has a finite number of sheets over $\Omega$.