Global branching of solutions to ODEs and integrability

Rod Halburd

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arXiv · 2502.15103

Global branching of solutions to ODEs and integrability

Rod Halburd

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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 $\Omega$.

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Rod Halburd. 2025-02-20. Global branching of solutions to ODEs and integrability. https://arxiv.org/abs/2502.15103

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