Abstract
This chapter contains some definitions and properties on the first integrals of ordinary differential equations. We will prove the local integrability of differential systems around a regular point, and the global existence and regularity of the first integrals of planar smooth differential systems in the canonical regions. As a preliminary application of the integrability theory, we use it to solve linear and quasilinear partial differential equations. Lastly, we introduce a few fundamental results on Lax pairs for finding the first integrals of ordinary differential equations.
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