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

TL;DR

This paper investigates special rational vector fields that are Liouvillian integrable but cannot be integrated using coefficients solely from the base field, providing explicit examples in three dimensions.

Contribution

It characterizes and constructs explicit examples of exceptional vector fields where the algebraic extension is necessary for Liouvillian integrability.

Findings

Existence of exceptional vector fields in dimension three.

Explicit constructions of such vector fields.

Characterization of when algebraic extensions are essential.

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.