Integrability of Lie systems through Riccati equations

TL;DR

This paper presents a geometric method to analyze integrability conditions for Riccati equations and Lie systems, offering a unified framework that can be extended to other Lie systems and aids in understanding their linearisability.

Contribution

It introduces a novel geometric approach to derive integrability conditions for Riccati equations, generalizable to all Lie systems, and clarifies their linearisability properties.

Findings

Unified geometric framework for Riccati integrability

Extension of methods to general Lie systems

Insights into Riccati linearisability

Abstract

Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyse a geometric method to construct integrability conditions for Riccati equations following these approaches. This approach provides us with a unified geometrical viewpoint that allows us to analyse some previous works on the topic and explain new properties. Moreover, this new approach can be straightforwardly generalised to describe integrability conditions for any Lie system. Finally, we show the usefulness of our treatment in order to study the problem of the linearisability of Riccati equations.