= Solution
For axisymmetric disturbances, $k_y=0$, the <wavevector> is fixed, so there is no winding/swing mechanism. Take $k_z\ne0$ and set $\chi=k_z^2/k^2$. Eliminating <pressure> with incompressibility gives
$$
\dot v_x=2\Omega\chi v_y-\nu k^2v_x,\qquad\dot v_y=-2(\Omega-A)v_x-\nu k^2v_y,\qquad v_z=-\frac{k_x}{k_z}v_x.
$$
The <axisymmetric incompressible epicyclic wave> has inviscid frequency
$$
\omega^2=\kappa_r^2\chi,\qquad\kappa_r^2=4\Omega(\Omega-A),
$$
where $\kappa_r$ is the <radial epicyclic frequency>.
In a Keplerian flow $A=3\Omega/4$ and $\kappa_r=\Omega$. The axisymmetric disturbances therefore execute stable inertial/epicyclic oscillations, with viscous <amplitude> decay $e^{-\nu k^2t}$. Rotation opposes sustained radial displacement through angular-momentum conservation. There can be bounded exchanges of <kinetic energy> between components, but no large Reynolds-number-dependent amplification of this axisymmetric family. Indeed, with $w=v_x/\sqrt\chi$, inviscid motion conserves
$$
(\Omega-A)|w|^2+\Omega|v_y|^2.
$$
Since the physical velocity norm is $|w|^2+|v_y|^2$, Keplerian <energy> amplification is at most $\Omega/(\Omega-A)=4$; <viscosity> can only reduce that bound. A pure initial azimuthal disturbance can attain the inviscid factor four after a quarter epicyclic period. By contrast, the two-dimensional nonaxisymmetric <Orr mechanism> remains available in a Keplerian flow and has the $\mathrm{Re}^{2/3}$ gain derived above.
In non-rotating shear, $\Omega=0$ while $A$ remains nonzero. The axisymmetric solution is
$$
v_x=C_xe^{-\nu k^2\tau},\qquad v_y=(C_y+2A\tau C_x)e^{-\nu k^2\tau},\qquad\tau=t-t_0.
$$
Cross-stream motion transports the basic shear and generates an azimuthal streak: this is the <lift-up effect>. Inviscid azimuthal velocity grows linearly and its <energy> quadratically, rather than oscillating. With <viscosity>, the gain for a suitable initial cross-stream disturbance scales as $[A/(\nu k^2)]^2$ at large <Reynolds number>, appreciably stronger than the two-dimensional $\mathrm{Re}^{2/3}$ scaling. For instance, $C_y=0$ and fixed $\chi>0$ give the leading optimized gain $4\chi A^2/(e^2\nu^2k^4)$ at $\tau\simeq1/(\nu k^2)$. Neither mechanism is an exponentially growing hydrodynamic <normal mode>. \b[Rotation bounds axisymmetric lift-up in <Keplerian shear>; without rotation, lift-up gives strong algebraic transient growth.] If $k_z=0$ as well as $k_y=0$, incompressibility forces $v_x=0$ and removes this coupling altogether.
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