= Solution
The spatial <reflection> $x\mapsto-x$ acts on the two complex <amplitudes> as $(A,B)\mapsto(A,-B)$, because the first spatial <eigenfunction> is even and the second is odd. The first equation must therefore be even in $B,\overline B$, and the second odd. For a nonresonant double <Hopf bifurcation>, the temporal <phases> give independent actions $A\mapsto e^{i\varphi_1}A$, $B\mapsto e^{i\varphi_2}B$ in the cubic <normal form of a dynamical system>. A cubic <monomial> in the first equation must have net $A$ <phase> weight one and net $B$ <phase> weight zero. Only $A|A|^2,A|B|^2$ satisfy this; the second equation similarly has $B|B|^2,B|A|^2$. Parametrizing the independent cross-saturation <coefficients> gives
$$
\begin{aligned}
\dot A&=\nu_1A-\alpha_1(|A|^2+|B|^2)A-\beta_1|B|^2A,\\
\dot B&=\nu_2B-\alpha_2(|A|^2+|B|^2)B-\beta_2|A|^2B.
\end{aligned}
$$
All <coefficients> can be complex: their <real parts> determine <amplitude> growth and their <imaginary parts> frequency shifts. These are cubic truncations of a family of reduced vector fields on the four-real-dimensional <extended centre manifold>, with the two unfolding parameters entering $\nu_1,\nu_2$.
The nonresonance assumption must be made explicit. Merely having $\omega_1\ne\omega_2$ does not imply this form: for $\omega_1=2\omega_2$, the quadratic terms $B^2$ in $\dot A$ and $A\overline B$ in $\dot B$ satisfy both temporal <resonance> and spatial parity. Thus the initial form is the generic nonresonant double-Hopf form, rather than a statement for every possible unequal frequency pair.
At the one-to-one <reflection-symmetric one-to-one Hopf resonance>, only the common temporal <phase> action remains. Cubic terms must then have total <phase> weight one. Spatial <reflection> retains, in the first equation, the three terms $A|A|^2,A|B|^2,B^2\overline A$; the second retains $B|B|^2,B|A|^2,A^2\overline B$. Common <phase> covariance excludes all quadratic terms. The additional resonant terms are consequently $-\gamma_1B^2\overline A$ and $-\gamma_2A^2\overline B$, with independent complex <coefficients>. This is the <reflection-symmetric one-to-one Hopf resonance>.
In the specified equal-coefficient specialization, put $r=R^2$. A nonzero pure-$A$ periodic branch exists for $\operatorname{Re}\nu_1>0$ and has
$$
r=\frac{\operatorname{Re}\nu_1}{\alpha_R},\qquad
\Omega=\operatorname{Im}\nu_1-\alpha_Ir.
$$
Its radial <eigenvalue> is $-2\alpha_Rr<0$, and its <phase> is neutral. For a transverse $B$ perturbation the linear equation is
$$
\dot B=[\nu_2-(\alpha+\beta)r]B-\beta r e^{2i\Omega t}\overline B.
$$
Its time dependence can be removed exactly by $B=C e^{i\Omega t}$. Using $\nu_1-\alpha r=i\Omega$ and $\nu_2=\nu_1-d$ gives
$$
\boxed{\dot C=-(d+\beta r)C-\beta r\overline C.}
$$
Writing $C=u+iv$, $d=d_R+id_I$, $\beta=\beta_R+i\beta_I$ yields the constant real <stability matrix>
$$
M(r)=\begin{pmatrix}-d_R-2\beta_Rr&d_I\\-d_I-2\beta_Ir&-d_R\end{pmatrix}.
$$
Let $Q=\operatorname{Re}(\beta\overline d)=\beta_Rd_R+\beta_Id_I$. The transverse <characteristic polynomial> is
$$
\boxed{\lambda^2+2(d_R+\beta_Rr)\lambda+|d|^2+2rQ=0.}
$$
Both transverse <eigenvalues> have negative <real part> exactly when
$$
d_R+\beta_Rr>0,\qquad |d|^2+2rQ>0.
$$
These hold at $r=0$ because $d_R>0$. Thus a sufficiently small pure branch is transversely stable; it is stable modulo its neutral temporal <phase> until one of the two inequalities fails. The <secondary steady and Hopf thresholds of a resonant pure mode> are derived in the two parts below. They refer to accessible points on the pure branch; in a multi-parameter problem $d$ may also vary, so a proposed parameter path must actually cross the corresponding surface.
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