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The axisymmetric magnetic mode spans a one-dimensional real representation of axial rotations, so its signed amplitude is a real variable $B$. An equatorial dipole has two real horizontal components. Rotation through an angle $\vartheta$ rotates that pair; encoding the components by one complex amplitude gives $A\mapsto e^{i\vartheta }A$, and rotation through $\pi$ gives $A\mapsto -A$. The velocity mode has the same rotational representation, so write it as $V\mapsto e^{i\vartheta }V$. Magnetic reversal acts asThus the two relevant symmetries are axial rotation and magnetic reversal. Reflection symmetry has not been assumed, so complex coupling coefficients are permitted.

The equations for $B,A$ must be linear in the magnetic variables at fixed velocity. Combining this with rotational equivariance, their general local form iswhere $f$ is real valued and $g,h,j,\ell$ are complex valued. These functions can be understood through their local Taylor expansions; the symmetry argument does not require a convergent infinite series. Reality of $\dot{B}$ explains its conjugate pairing.

For the velocity equation, a monomial $B^p{A}^q(A^{\ast }{)}^r{V}^s(V^{\ast }{)}^t$ must have the same rotational weight as $V$ and be even under magnetic reversal. Its selection rules areA formal sum of all such monomials with complex coefficients is the most general local velocity evolution. At degree one there is $V$; at degree two there is $AB$; at degree three there are $V|V{|}^2,V|A{|}^2,V^{\ast }{A}^2,V{B}^2$. Expanding the magnetic equations similarly gives the equivariant three-mode dynamo normal formThe ${\mu }_i,{\omega }_i,e$ are real, and the remaining coefficients are complex. The minus signs on cubic coefficients are a convention, not an assumption about their signs. Terms such as $B^3$ and $A|A{|}^2$ are forbidden in the magnetic equations by magnetic linearity, even though symmetry alone would allow some of them.

For the normalization of nonzero dynamo coupling coefficients, assume $c_1{c}_3\ne 0$. Make the variable changesThe transformed coefficients are ${\tilde{c}}_1=qv{e}^{i({\chi }_V-{\chi }_A)}{c}_1=1$ and ${\tilde{c}}_3=(q/v)e^{i({\chi }_A-{\chi }_V)}{c}_3$, of modulus one. ThusThe remaining phase is genuine; the two complex coefficients cannot both generally be made positive real. A zero coupling cannot be normalized to one, so the printed normalization presupposes this generic nonzero-coupling case.

Now impose the specified simplification, in the normalized variables, and write $A=r{e}^{i\theta }$, $V=s{e}^{i\varphi }$ with $r,s>0$ and signed real $B$. Put $\psi =\varphi -\theta$. Separating real and imaginary parts givesConstant moduli require the first three right sides to vanish. Since $rs\ne 0$, the first equation also fixes $\cos \psi$, so a continuously evolving phase difference must be constant. Hence the last right side must vanish as well. Both phases then advance at one common frequency, giving a relative equilibrium. These four algebraic conditions and the displayed common-frequency equations govern all the requested nonzero constant-modulus solutions, without needing to solve them.

The pure-velocity branch isThis presumes $d_4\ne 0$; the usual saturated branch has ${\mu }_3,d_4>0$. Near it, set $A=e^{i({\omega }_{3}t+{\varphi }_0)}(u+iv)$, $B=b$, and define $\Delta ={\omega }_2-{\omega }_3$ and $\kappa ={\mu }_1-e{s}_0^2$. Magnetic feedback in the velocity equation is quadratic, so the first magnetic linearization isA branch of the stated relative equilibria has a zero growth rate in this rotating frame. Its determinant therefore gives the magnetic onset on a rotating velocity branch conditionFor ${\mu }_2^2+{\Delta }^2\ne 0$, the zero-mode relation is $u+iv=-c_2{s}_{0}b/({\mu }_2+i\Delta )$, so a nonzero $b$ gives a nonzero $A$ when $c_2\ne 0$. With a transverse eigenvalue crossing and nondegenerate nonlinear saturation this is the generic magnetic bifurcation surface; magnetic reversal pairs its two signs. Degenerate cases require separate treatment. For example, if ${\mu }_2=\Delta =0$ and $c_2\ne 0$, determinant zero is automatic but the stationary $u$ equation forces $b=0$, so that zero mode does not establish the requested branch with both magnetic amplitudes nonzero.