Algebraic nonsolvability of the problem of existence of holomorphic first integrals

Abstract

We prove that the existence of a holomorphic first integral of an analytic differential equation is an algebraically nonsolvable problem. Moreover, we prove a perturbation lemma showing that a nilpotent singularity of a differential equation in the plane cannont have a holomorphic first integral independently of the prolongation of anynth-order jet. We give an application to the case of nilpotent centers.