Phase locking of a resonant hexagonal triad ($\dot{\Phi }=-\alpha \left(\frac{r_2{r}_3}{r_1}+\frac{r_3{r}_1}{r_2}+\frac{r_1{r}_2}{r_3}\right)\sin \Phi$)

For $A_i=r_i{e}^{i{\varphi }_i}$ with all $r_i>0$, the translation-invariant total phase is $\Phi ={\varphi }_1+{\varphi }_2+{\varphi }_3$. With $\alpha >0$, the displayed equation makes $\Phi =0$ stable and $\Phi =\pi$ unstable. A spatial translation can remove two phases, so the stable locked states have all amplitudes positive real in a suitable origin. This is a statement modulo translation, not that every translated representative is real; the zero solution and rolls belong to boundary strata.