= Equivariant three-mode dynamo normal form
{title2=$B\in\mathbb R,\quad A,V\in\mathbb C$}
An axisymmetric magnetic mode is unchanged by axial rotations, while an equatorial magnetic mode and a corresponding velocity mode transform as $A,V\mapsto e^{i\theta}(A,V)$. Magnetic reversal sends $(B,A,V)$ to $(-B,-A,V)$. For magnetic evolution linear in magnetic amplitudes, rotational <equivariance> requires $\dot B=Bf(|V|^2)+2\operatorname{Re}[A^*Vg(|V|^2)]$ and $\dot A=Ah(|V|^2)+BVj(|V|^2)+A^*V^2\ell(|V|^2)$, with $f$ real and the other functions complex. Velocity monomials $B^pA^q(A^*)^rV^s(V^*)^t$ must have even $p+q+r$ and weight $q-r+s-t=1$. These conditions enumerate the cubic <normal form> without assuming reflection symmetry or nonlinear magnetic self-interactions.
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