Formal First Integrals Along Solutions of Differential Systems I

We consider an analytic vector field $\dot{x}= X(x)$ and study, via a variational approach, whether it may possess analytic first integrals. We assume one solution $\Gamma$ is known and we study the successive variational equations along $\Gamma$. Constructions in the paper by Morales, Ramis and Simó show that Taylor expansions coefficients of first integrals appear as rational solutions of the dual linearized variational equations. We show that they also satisfy linear "filter" conditions. Using this, we adapt the algorithms from Barkatou 99 and van Hoeij and Weil 97 to design new ones optimized to this effect and demosntrate their use. Part of this work stems from the first author's Ph. D. thesis.