 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 $-d{r}^2\pm \sqrt{{\delta }^2{r}^4+r^2}$, each twice, radial eigenvalue $-2\mu$, and one neutral translation. A hexagon with $\mu =N{h}^2-h$ has real-amplitude eigenvalues $h-2N{h}^2$ and $-2h+2d{h}^2\pm 2i\sqrt{3}\delta h^2$; 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 $\delta \ne 0$, the upper positive hexagon branch reaches a Hopf bifurcation at $h=1/d$, $\mu =(2{\nu }_1+{\nu }_2)/d^2$.