 Signed cyclic three-mode amplitude equations

Independent sign changes of three amplitudes and a cyclic coordinate permutation force each cubic component to be its own amplitude times a cyclic permutation of one quadratic polynomial. One convenient parametrization is $\dot{x}=x(\mu -aX-c{y}^2+e{z}^2)$ with cyclic companions and $X=x^2+y^2+z^2$. The axial and diagonal isotropy types give primary branches; two-coordinate branches can emerge in secondary pitchfork bifurcations when cross-couplings change sign.