= Solution
Substitute the axisymmetric <Fourier mode> into the linearized equations. With hats denoting amplitudes,
$$
\begin{aligned}
-i\omega\hat v_x-2\Omega\hat v_y&=-ik_xc^2\hat\delta,\\
-i\omega\hat v_y+\frac\Omega2\hat v_x&=0,\\
-i\omega\hat v_z&=-ik_zc^2\hat\delta,\\
\omega\hat\delta&=k_x\hat v_x+k_z\hat v_z.
\end{aligned}
$$
For a nondegenerate nonzero frequency, eliminate the azimuthal velocity and use
$$
(\omega^2-\Omega^2)\hat v_x=\omega k_xc^2\hat\delta,\qquad \omega\hat v_z=k_zc^2\hat\delta.
$$
Continuity then gives
$$
\omega^2(\omega^2-\Omega^2)=c^2\left[k_x^2\omega^2+k_z^2(\omega^2-\Omega^2)\right].
$$
Equivalently, taking the <determinant> of the original four amplitude equations avoids division by zero and gives the full <dispersion relation>
$$
\boxed{\omega^4-(\Omega^2+c^2k^2)\omega^2+\Omega^2c^2k_z^2=0,\qquad k^2=k_x^2+k_z^2.}
$$
The two squared frequencies are
$$
\omega_\pm^2=\frac12\left[\Omega^2+c^2k^2\pm\sqrt{(\Omega^2+c^2k^2)^2-4\Omega^2c^2k_z^2}\right].
$$
The <discriminant> is $(\Omega^2-c^2k^2)^2+4\Omega^2c^2k_x^2\ge0$. For $c^2>0$, both squared frequencies are nonnegative. The high-frequency branch is predominantly acoustic and the low-frequency branch is an <inertial wave>. Zero-frequency limiting modes, including $k_z=0$, are retained by the determinant calculation.
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