= Solution
The <equivariant Hopf theorem> states that, under the usual nondegeneracy and transverse-crossing assumptions at an equivariant Hopf point, each complex axial isotropy subgroup of the spatial symmetry group combined with temporal phase produces a branch of periodic solutions with that spatiotemporal symmetry. Its common fixed-point space must have real dimension two, or complex dimension one. Restricting the Hopf <normal form of a dynamical system> to that plane gives a scalar complex <amplitude equation>; a nonzero real cubic coefficient provides a small periodic branch on the corresponding parameter side. The theorem supplies existence and symmetry, not stability.
Put $\zeta=e^{2\pi i/3}$ and write $T_\theta z=e^{i\theta}z$, so $T_\theta=\tau_{\theta/\omega_0}$. Three complex axial fixed lines are
$$
\begin{array}{c|c|c}
\text{type}&\text{isotropy generator}&\text{fixed line}\\\hline
\text{rotating wave}&T_{2\pi/3}\rho&(z,0)\\
\text{standing wave }+&m_x&(z,z)\\
\text{standing wave }-&T_\pi m_x&(z,-z).
\end{array}
$$
For the first generator, its <eigenvalues> are $1,\zeta^2$, so the fixed line is exactly $z_2=0$. Reflection supplies the conjugate line $z_1=0$, representing the opposite rotation direction. For the other two generators, swapping coordinates, with or without the common minus sign, gives the displayed one-complex-dimensional fixed spaces.
These subgroups are inequivalent: the rotating-wave spatial projection has order three, while the standing-wave projections have order two; the two reflection types differ by the central phase $T_\pi$, which conjugation cannot remove. Rotating the two standing lines by $\rho$ gives three orientations of each type, not new symmetry types. Therefore \b[there are three distinct generic periodic branch types: rotating waves and two inequivalent standing-wave families]. Their members have a period near $2\pi/\omega_0$, with the combined spatial–temporal operations determining their relative phase relations.
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