About algebraic integrability and non-integrability of ordinary differential equations

✍ Scribed by Andrzej J. Maciejewski

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 596 KB
• Volume: 9
• Category: Article
• ISSN: 0960-0779
• DOI: 10.1016/s0960-0779(97)00048-9

✦ Synopsis

Abstrael--In this paper I describe some effective methods that can be applied to the study of the problem of existence of a polynomial or rational first integral of an arbitrary system of ordinary differential equations with polynomial right hand sides. I present an extension of the method of compatible vector fields. Next, I show that for quadratic three-dimensional factorizable systems the existence of an invariant measure can be used to distinguish a family of co-dimensions in parameter space with an explicitly given first integral in •3. Finally. I demonstrate the Lagutinskii-Levelt method which, used together with some additional tools, gives the strongest results concerning nonintegrability.