Local Integrability of Differential Systems

Abstract

In Chap. 1 we proved that for \(k\in \mathbb N\cup \{\infty ,\omega \}\), an n-dimensional \(C^k\) autonomous differential system always has \(n-1\) functionally independent \(C^k\) first integrals in a neighborhood of a regular point, where the first integrals are independent of the independent variable of the system. This chapter will concentrate on the existence of analytic or formal first integrals of analytic differential systems in a neighborhood of a singularity, with an emphasis on the varieties and the existence of analytic normalizations of analytic integrable (or partially integrable) differential systems. We will also introduce the local theory of Darboux integrability of local analytic or formal differential systems.