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arXiv · 2512.22138

Liouvillian integrability of rational vector fields: The case of algebraic extensions

Colin Christopher·Chara Pantazi·Sebastian Walcher
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Abstract

As shown in a previous paper, whenever a rational vector field on $\mathbb C^n$, $n>2$, is Liouvillian integrable, then it admits a first integral obtained by two successive integrations from a one-form with coefficients in a finite algebraic extension $L$ of the rational function field $K$. In the present work we discuss and characterize exceptional vector fields in this class, for which -- by definition -- the choice $L=K$ is not possible. In particular we show that exceptional vector field exist, giving explicit constructions in dimension three.

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Colin Christopher, Chara Pantazi, Sebastian Walcher. 2025-12-11. Liouvillian integrability of rational vector fields: The case of algebraic extensions. https://arxiv.org/abs/2512.22138

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