About algebraic integrability and non-integrability of ordinary differential equations

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 1. Finally, I demonstrate the Lagutinskii-Levelt method which, used together with some additional tools, gives the strongest results concerning non-integrability.