Trivial Isochronous Centers in Odd Degrees: a Two--Branch Picture

Abstract

We revisit the characterization of \emph{trivial} isochronous centers for planar polynomial Hamiltonian systems in degrees $5$ and $7$ obtained by Braun--Llibre--Mereu, and we formalize two conclusions suggested by their method. First, a \emph{triangular family} yields trivial (indeed global) isochronous centers in every odd degree $n=2k-1\geq3$. Second, a genuinely different \emph{quadratic--shear} ($Q$) family appears exactly when $n\equiv3\pmod 4$, beginning at $n=7$, explaining the observed \textquotedblleft alternating\textquotedblright\ emergence of a second branch. For $n=9$ this second branch cannot occur by degree parity. Our statements rest on the structure of the degree--7 proof and the general triangular construction in the preprint, together with the standard isochrony characterization $\mathcal{H}=\tfrac{1}{2}(f_{1}^{2}+f_{2}^{2})$ with $\det Df\equiv1$.