= Roll and hexagon spectra without reflection symmetry
For the <rotational hexagon amplitude equations>, put $d=\nu_2-\nu_1$ and $N=\nu_1+2\nu_2$. A roll with $r^2=\mu/\nu_1$ has transverse <eigenvalues> $-dr^2\pm\sqrt{\delta^2r^4+r^2}$, each twice, radial <eigenvalue> $-2\mu$, and one neutral translation. A hexagon with $\mu=Nh^2-h$ has real-amplitude <eigenvalues> $h-2Nh^2$ and $-2h+2dh^2\pm2i\sqrt3\delta h^2$; its imaginary-amplitude <eigenvalues> are $-3h,0,0$. These follow by diagonalizing the cyclic <Jacobian matrix>. Stability is understood modulo translations. If $\nu_2>\nu_1>0$ and $\delta\ne0$, the upper positive hexagon branch reaches a <Hopf bifurcation> at $h=1/d$, $\mu=(2\nu_1+\nu_2)/d^2$.
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