A note on the Darboux theory of integrability of non-autonomous polynomial differential systems (Q2918662)

The authors give sufficient and necessary conditions for the existence of a Darboux first integral or a Darboux Jacobi multiplier for a polynomial vector field with some known algebraic invariant hypersurfaces. For the autonomous case, the classical result of Darboux is obtained as a corollary. For the non-autonomous case, the characterization improves a known criterium given by \textit{J. Llibre} and \textit{Ch. Pantazi} [J. Math. Phys. 50, No. 10, Paper No. 102705 (2009; Zbl 1283.34027)]. A possible generalization of the results presented in this work is via the results presented in [\textit{I. A. García} and \textit{J. Giné}, Appl. Math. Lett. 16, No. 7, 1137--1141 (2003; Zbl 1057.34002)].