TITLE:
Formal First Integrals Along Solutions of Differential Systems I

AUTHORS:
Ainhoa Aparicio-Monforte, Moulay Barkatou, Sergi Simon, Jacques-Arthur Weil.
E-mails: aparicio@risc.uni-linz.ac.at, moulay.barkatou@unilim.fr,
         sergi.simon@port.ac.uk, weil@unilim.fr

ABSTRACT:
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 study the successive variational
equations along $\Gamma$. Constructions by Morales-Ruiz, Ramis and Simo show
that Taylor expansion 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 by Barkatou, Van Hoeij and Weil to design new ones optimized to this
effect and demonstrate their use. Part of this work stems from the first
author's Ph.D. thesis (2010).