= Dihedral threefold Hopf normal form
For the $D_3$ action $\rho(z_1,z_2)=(\zeta^{-1}z_1,\zeta z_2)$, $m(z_1,z_2)=(z_2,z_1)$ and common phase multiplication, the fifth-order normal form has first component $z_1[\alpha+a|z_1|^2+b|z_2|^2+c|z_1|^4+d|z_1|^2|z_2|^2+e|z_2|^4]+f\bar z_1^2z_2^3$ and second component obtained by swapping indices. Here all coefficients may be complex. Its three primary symmetry types are a one-component rotating wave and the two standing-wave lines $z_1=z_2$ and $z_1=-z_2$, together with their group conjugates. The two standing amplitudes split at fifth order through $\operatorname{Re}f$.
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