The absence of periodic solutions for a system of differential equations (Q677693)

We consider a system of differential equations (1) \({dx\over dt}=y\), \({dy\over dt}=\sum^m_{k=0}f_k(x)y^k\), \(m>3\), where \(f_k(x)\), \(k=\overline{0,m}\), are analytic functions for all real \(x\), \(f_m(x)\not\equiv 0\). In [\textit{Ya. N. Shnejderman}, Differ. Integral'nye Uravn., Gor'kij, 24-28 (1983)] necessary and sufficient conditions were obtained for such a system to have a general semi-algebraic integral of a given degree. In [\textit{N. I. Avdonin}, Differ. Uravn. 4, 639-645 (1968; Zbl 0164.10501)] the sufficient conditions for acyclicity of the system (1) (i.e., the absence of closed trajectories) were found on the base of a generalized symmetry method. The question of the existence of limit cycles for (1) was also considered in [\textit{F. H. Sattarov}, Differ. Uravn. 10, 844-850 (1974; Zbl 0291.34021)] and [\textit{E. D. Zhitel'zejl}, Differ. Uravn. 11, 1702-1704 (1975; Zbl 0307.34026)].