Formal first integrals along solutions of differential systems I

We consider an analytic vector field ẋ = X(x) and study whether it may possess analytic first integrals via a variational approach. We assume that one solution Γ  is known and study the successive variational equations along Γ. Constructions of Morales-Ramis-Simo show that coefficients of the Taylor expansions of first integrals arise as rational solutions of the dual linearized variational equation. We show that they also satisfy linear “filter” conditions. Using this, we adapt the algorithms from to design algorithms optimized for this task and demonstrate their use. Part of this work stems from the first author’s PhD thesis .