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First there is a necessary coordinate qualification. With the literal $z$ action obtained in part (c), the temporal weights are $-1,+1$ and reflection is conjugation followed by exchange. A linear term with $(\mu +i\omega )$ in both equations fails reflection equivariance for $\omega \ne 0$, since conjugation changes the sign of $i\omega$. The proposed first-component monomial ${\bar{z}}_1{z}_2^2$ also has temporal weight $+3$, rather than $-1$. Thus the displayed equations cannot be proved in those literal coordinates. They are the correct dihedral fourfold Hopf normal form in the common-phase Hopf coordinates $(q_1,q_2)=(z_1,{\bar{z}}_2)$.

In these coordinates the temporal circle imposes total phase weight one on every vector-field monomial. There are no quadratic monomials with that weight. A cubic monomial has two unbarred factors and one barred factor. For the first component, rotation by $\rho$ must multiply the monomial by $i$. Inspecting the six phase-compatible monomials leaves preciselyThe excluded three monomials multiply by $-i$ instead. Reflection swaps the components, so their coefficients must be the same after swapping indices. The linear part is a scalar complex multiple by absolute irreducibility. Orient the Hopf parameter to write it as $\mu +i\omega$, and name the cubic coefficients $-a,-b,-c$, where $a,b,c$ are generally complex. This provesThese are cubic amplitude normal forms of a dynamical system, after the usual symmetry-compatible Hopf normalization. Returning to literal $z$ coordinates conjugates the second equation: its linear coefficient is $\mu -i\omega$, its cubic coefficients are $\bar{a},\bar{b},\bar{c}$, and the first coupling is $-c{\bar{z}}_1{\bar{z}}_2^2$. That corrected form respects every printed generator.

On the three complex axial isotropy subgroup fixed spaces, substitute $q_1=R{e}^{i\Omega t}$ and $q_2=0,q_1,i{q}_1$. The radial equations are $\dot{R}=R(\mu -A_j{R}^2)$, withTherefore the leading component amplitudes areEach expression applies on the parameter side where it is positive, with its denominator nonzero for a nondegenerate branch. The frequencies are ${\Omega }_j=\omega -\mathrm{Im}(a_j)R_j^2$, with $a_j=a,a+b+c,a+b-c$. The denominators are generically distinct, so coexisting branches generically have different component amplitudes. If the amplitude is instead measured by the original real-space norm, the identity $|w_1{|}^2+|w_2{|}^2=2(|q_1{|}^2+|q_2{|}^2)$ gives squared amplitudes $2R_R^2$, $4R_A^2$, and $4R_D^2$, again generically distinct. Equalities require special coefficient relations, rather than symmetry forcing them.