Case I. $A^*_{33} - A^*_{11}/2 \ne 0$, A^*₁₃ = 0.

Whenever $A^*_{33} - A^*_{11}/2 \ne 0$ and $A^*_{13} \to 0$, we find easily that $\theta^*_- \to 0$, while $\theta^*_+ \to \pi/2$. In this case, v₁ and v₃ are themselves the eigenvectors, while the eigenvalues are proportional to A^*₁₁ and A^*₃₃. In the quasi-isotropic limit, A^*₁₃ can vanish only if $\beta_1 -\beta_3 =0$, in which case A^*₃₃ also does not depend on fluid properties. For media differing significantly from the quasi-isotropic limit, A^*₁₃ could vanish for some physically interesting situations, but the resulting physical constraints are too special (and complicated) for us to consider them further here.