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

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 $n=2k-1\ge 3$. Second, a genuinely different \emph{quadratic–shear} ($Q$) family appears exactly when $n\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$, 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 $\mathcal{H}=\frac{1}{2}(f_1^2+f_2^2)$ with $\det Df\equiv 1$.