Dynamics of polynomial generalized Liénard system near the origin and infinity

The paper’s Theorem 2.2 states exactly the monodromy alternatives (M1) p(r+1)<q(p+1) with p,r odd and cr<0, and (M2) p(r+1)=q(p+1) with p,r odd and cr<−ĉ (ĉ defined in Table 1), and gives the center criterion: under monodromy, O is a center iff the system F(x)=F(z), G(x)=G(z) has a unique z(x) with z(0)=0, z′(0)<0. These match the candidate solution verbatim, including the ap=−1 normalization, Newton polygon/quasi-homogeneous blow-ups, and Cherkas-based return-map characterization. The paper’s proof sketch uses the same machinery (blow-ups guided by the Newton polygon; Cherkas for the first return), so the two arguments are substantially the same and yield identical conditions, including the borderline threshold.