  The question of how Algebra can be used to solve dynamical systems and
characterize chaos was first posed in a fertile mathematical context by Ziglin,
Morales, Ramis and Sim\'o using differential Galois theory. Their study was
aimed at first-order, later higher-order, variational equations of Hamiltonian
systems. Recent work by this author formalized a compact yet comprehensive
expression of higher-order variationals as one infinite linear system, thereby
simplifying the approach. More importantly, the dual of this linear system
contains all information relevant to first integrals, regardless of whether the
original system is Hamiltonian. This applicability to formal calculation of
conserved quantities is the centerpiece of this paper, following an
introduction to the requisite context. Three important examples, namely
particular cases of Dixon's system, the SIR epidemiological model with vital
dynamics and the Van der Pol oscillator, are tackled, and explicit convergent
first integrals are provided for the first two.

Dynamical Systems