Weierstrass integrability of differential equations

TL;DR

This paper introduces the concept of Weierstrass integrability for differential equations, characterizes systems with elementary or Liouvillian integrals, and explores the relationship between Weierstrass and Liouvillian integrability, including non-Liouvillian cases.

Contribution

It defines Weierstrass integrability and determines its relation to Liouvillian integrability, expanding the classification of integrable differential systems.

Findings

Characterization of systems with elementary or Liouvillian first integrals

Introduction of Weierstrass integrability as a new class

Identification of non-Liouvillian integrable systems within this class

Abstract

The integrability problem consists in finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define {\it Weierstrass integrability} and we determine which Weierstrass integrable systems are Liouvillian integrable. Inside this new class of integrable systems there are non--Liouvillian integrable systems.