Integrability and Limit Cycles via First Integrals

In many problems appearing in applied mathematics the nonlinear ordinary differential systems, as physics, chemist, economics, etc., if we have a system on manifold of dimension, two them having first integral, then its phase portrait is completely determined. While existence integrals for systems manifolds dimension higher than allows to reduce space dimensions independent have. Hence, know important, but following question appears: Given system, how it has integral? The symmetries force integrals. This paper main objectives. First, study compute polynomial using so-called Darboux theory integrability. Furthermore, second, show use finding limit cycles piecewise systems.