equation 1:
$-i\omega\underline \xi - 2i(\underline k \cdot \underline \Omega )\underline u = 0$
with:
$\underline \xi = i \underline k \times \underline u$
$\underline k \cdot \underline u = 0$
taking:
$i\underline k \times \text{(1)}$
derive:
$\omega = \pm 2\frac{(\underline k \cdot \underline \Omega )}{|\underline k|}$
I can get pretty close and can post my working if required - but I don't do anything complicated - any help = many thanks