= Solution
For an <axisymmetric vertical mode of a shearing sheet> with real $k\ne0$, the two divergence constraints give $ikv_z=ikb_z=0$. The vertical momentum equation then yields $ikp=0$. Thus \b[all three amplitudes vanish]:
$$
\boxed{v_z=b_z=p=0.}
$$
Define the magnetic amplitudes $a_i=b_i/\sqrt{4\pi\rho_0}$ and the signed vertical <Alfvén velocity> $v_A=B_0/\sqrt{4\pi\rho_0}$. Write $A=k^2v_A^2$. The remaining equations for the <normal mode> are
$$
\begin{aligned}
\sigma v_x-2\Omega v_y&=-N^2\vartheta+ikv_Aa_x,\\
\sigma v_y+\frac12\Omega v_x&=ikv_Aa_y,\\
\sigma a_x&=ikv_Av_x,\\
\sigma a_y&=-\frac32\Omega a_x+ikv_Av_y,\\
\sigma\vartheta&=v_x.
\end{aligned}
$$
For a growing or oscillatory mode with $\sigma\ne0$, eliminating $a_x,a_y,\vartheta$ gives
$$
(\sigma^2+A+N^2)v_x-2\Omega\sigma v_y=0,
$$
$$
\sigma(\sigma^2+A)v_y+\left(\frac12\Omega\sigma^2-\frac32\Omega A\right)v_x=0.
$$
A nonzero <velocity> requires the determinant to vanish. After removing its factor $\sigma$, \b[the dynamical <dispersion relation>] is
$$
\boxed{\sigma^4+(2A+N^2+\Omega^2)\sigma^2+A(A+N^2-3\Omega^2)=0.}
$$
This is the <radially stratified magnetorotational dispersion relation>, with $A=k^2B_0^2/(4\pi\rho_0)$. Keeping the original five-amplitude system instead gives characteristic polynomial $\sigma$ times this quartic. There is also a stationary balanced <normal mode>; division by $\sigma$ excludes it but loses no exponentially growing mode. At $k=0$ the divergence argument for vanishing vertical components does not apply, so that spatially uniform case must be treated separately.
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