  The analytic galactic wind model derived by Chevalier and Clegg in 1985
(CC85) assumes $\textit{uniform}$ energy and mass-injection within the
starburst galaxy nucleus. However, the structure of nuclear star clusters,
bulges, and star-forming knots are non-uniform. We generalize to cases with
spherically-symmetric energy/mass injection that scale as $r^{-\Delta}$ within
the starburst volume $R$, providing solutions for $\Delta = 0$, 1/2, 1, 3/2,
and 2. In marked contrast with the CC85 model ($\Delta=0$), which predicts zero
velocity at the center, for a singular isothermal sphere profile ($\Delta=2$),
we find that the flow maintains a $\textit{constant}$ Mach number of
$\mathcal{M}=\sqrt{3/5} \simeq 0.77$ throughout the volume. The fast interior
flow can be written as $v_{r < R} = (\dot{E}_T/3\dot{M}_T)^{1/2} \simeq 0.41 \,
v_\infty$, where $v_\infty$ is the asymptotic velocity, and $\dot{E}_T$ and
$\dot{M}_T$ are the total energy and mass injection rates. For $v_\infty \simeq
2000 \, \mathrm{km \, s^{-1}}$, $v_{r<R} \simeq 820 \, \mathrm{km\, s^{-1}}$
throughout the wind-driving region. The temperature and density profiles of the
non-uniform models may be important for interpreting spatially-resolved maps of
starburst nuclei. We compute velocity resolved spectra to contrast the
$\Delta=0$ (CC85) and $\Delta=2$ models. Next generation X-ray space telescopes
such as XRISM may assess these kinematic predictions.
