Formal Weierstrass integrability for a Liénard differential system (Q2660475)

In the present paper, the authors study the following real Liénard differential system \[ x'=y,\quad y'=-(\zeta x^{3}+\alpha x+\beta)y-(\epsilon x^{3}+\sigma x). \tag{\(*\)} \] The cases \(\zeta \neq 0\) and \(\zeta =0\) with \(\alpha\neq 0\) are studied here, since the case \(\zeta = \alpha =0\) is studied earlier in [\textit{J. Giné} and \textit{J. Llibre}, Electron. J. Qual. Theory Differ. Equ. 2020, Paper No. 1, 16 p. (2020; Zbl 1449.34001)]. The authors apply the techniques developed in [\textit{J. Giné} and \textit{J. Llibre}, Int. J. Bifurcation Chaos Appl. Sci. Eng. 30, No. 4, Article ID 2050064, 7 p. (2020; Zbl 1446.34001)] and [loc. cit.] to equation (\(*\)) for finding non-Liouvillian integrable systems.