On elementary integrability of rational vector fields

TL;DR

This paper characterizes complex rational vector fields with elementary first integrals, especially the exceptional cases beyond Darboux integrability, by analyzing algebraic extension fields and providing construction criteria.

Contribution

It provides a complete characterization and construction of exceptional elementary integrable rational vector fields in two dimensions, extending the understanding beyond Darboux integrability.

Findings

Characterization of algebraic extension fields for exceptional cases

Construction methods for all exceptional vector fields

Criteria limiting the degree of the extension field

Abstract

We consider complex rational vector fields that admit a first integral whose logarithmic derivative lies in a finite extension of the rational function field $K$. In view of the Prelle-Singer theorem, these are the rational vector fields that admit an elementary first integral. Elementary integrable vector fields which are not Darboux integrable -- thus the extension field is necessarily a proper extension of $K$ -- may be called exceptional by an observation in an earlier paper by Christopher et al. For dimension two we characterize all possible algebraic extension fields underlying the exceptional cases, provide a construction of all exceptional vector fields, and obtain some criteria that restrict the degree of $L$.