= Solution
Let $k=|k_x|$, $c_s^2=(dP/d\Sigma)_0$ and $\kappa^2=2\Omega(2\Omega-S)$. Linearize the <barotropic closure of a razor-thin disk> and <shearing sheet> about the given state. Write the radial and azimuthal <velocity> perturbations as $u$ and $v$. <Axisymmetry> removes advection by the background shear, giving
$$
-i\omega\Sigma_1+ik_x\Sigma_0u=0,\qquad
-i\omega u-2\Omega v=-ik_x\left(\Phi_1+\frac{c_s^2\Sigma_1}{\Sigma_0}\right),\qquad
-i\omega v+(2\Omega-S)u=0.
$$
The softened potential is $\Phi_1=-2\pi G\Sigma_1e^{-k\epsilon}/k$. Eliminating the <velocity> components for the compressive branch gives the <softened Toomre dispersion relation>
$$
\boxed{\omega^2=\kappa^2-2\pi G\Sigma_0k e^{-k\epsilon}+c_s^2k^2.}
$$
The Coriolis/shear combination supplies the radial epicyclic restoring term, <pressure> supplies the short-wave term, and <self-gravity> lowers the squared frequency. The full three-variable determinant also has a stationary <axisymmetric geostrophic mode>, with $u=0$ and azimuthal flow balancing the pressure-plus-gravity gradient. The displayed relation describes the density-wave pair; eliminating by division by $\omega$ must not silently deny that stationary mode.
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