First integrals and phase portraits of planar polynomial differential cubic systems with invariant straight lines of total multiplicity eight

摘要

In [C. Bujac, J. Llibre, N. Vulpe, Qual. Theory Dyn. Syst. 15(2016), 327–348] allfirst integrals and phase portraits were constructed for the family of cubic differentialsystems with the maximum number of invariant straight lines, i.e. 9 (considered withtheir multiplicities). Here we continue this investigation for systems with invariantstraight lines of total multiplicity eight. For such systems the classification according tothe configurations of invariant lines in terms of affine invariant polynomials was donein [C. Bujac, Bul. Acad. S, tiint,e Repub. Mold. Mat. 75(2014), 102–105], [C. Bujac, N. Vulpe,J. Math. Anal. Appl. 423(2015), 1025–1080], [C. Bujac, N. Vulpe, Qual. Theory Dyn. Syst.14(2015), 109–137], [C. Bujac, N. Vulpe, Electron. J. Qual. Theory Differ. Equ. 2015, No. 74,1–38], [C. Bujac, N. Vulpe, Qual. Theory Dyn. Syst. 16(2017), 1–30] and all possible 51configurations were constructed. In this article we prove that all systems in this classare integrable. For each one of the 51 such classes we compute the corresponding firstintegral and we draw the corresponding phase portrait.