= Solution
Without the threefold rotation, a complex linear map need only commute with coordinate swapping and common phase. Commuting with $m_x$ makes its two diagonal entries equal and its two off-diagonal entries equal; common phase still forbids conjugate-linear terms. Thus the additional leading coupling is
$$
\boxed{\dot z_1=\alpha z_1+\eta z_2+\cdots,\qquad
\dot z_2=\alpha z_2+\eta z_1+\cdots,}
$$
where $\eta$ is a small complex symmetry-breaking coefficient and a small common diagonal shift can be absorbed into $\alpha$. The old rotation forbade these off-diagonal entries because the two coordinates have different rotation weights. Further symmetry-breaking cubic and quintic terms are also allowed; retaining only the new linear terms describes the leading onset splitting.
In symmetric and antisymmetric coordinates $z_+=(z_1+z_2)/\sqrt2$, $z_-=(z_1-z_2)/\sqrt2$, the <eigenvalues> are $\alpha+\eta$ and $\alpha-\eta$. With $\alpha=\mu+i\omega_0$, the two primary Hopf thresholds and frequencies are therefore
$$
\boxed{\mu_+=-\operatorname{Re}\eta,\quad\mu_-=+\operatorname{Re}\eta,\qquad
\omega_+=\omega_0+\operatorname{Im}\eta,\quad
\omega_-=\omega_0-\operatorname{Im}\eta.}
$$
The reflection-preserving standing line $z_1=z_2$ remains invariant. The line $z_1=-z_2$ also remains invariant because it is fixed by reflection combined with phase $\pi$. Their generic periodic branches survive, now starting at separate thresholds when $\operatorname{Re}\eta\ne0$. If that real part vanishes, threshold coincidence is a nongeneric additional condition; frequency splitting may still occur.
The pure rotating-wave axes are not invariant: at $z_2=0$ the second equation has $\dot z_2=\eta z_1$. Hence \b[the old pure rotating-wave branches cease to be primary one-component branches from the origin]. At fixed nonzero symmetry breaking, a simple Hopf onset has just the appropriate symmetric or antisymmetric mode. Away from the degenerate onset, normally hyperbolic rotating-wave <periodic orbits> may persist as mixed-mode periodic states with both components nonzero, paired by reflection. In common unfoldings such a pair is reached through a secondary reflection-breaking <pitchfork bifurcation> of <periodic orbits> from a standing branch. Its precise attachment, any folds, and stability depend on the nonlinear coefficients and cannot be fixed by symmetry alone.
The three rotationally conjugate orientations within each original standing family are no longer equivalent. Only the lines aligned with the surviving reflection symmetries are forced primary branches; other members can deform, pin to preferred orientations, or connect through secondary bifurcations. The generic <bifurcation diagram> thus replaces the coincident multi-branch Hopf onset by two separated simple Hopf onsets, removes the pure traveling-wave invariance, and unfolds the rotational degeneracies. This is the <reflection-preserving splitting of a dihedral Hopf bifurcation>; it does not by itself prescribe a unique global diagram.
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