The analytic galactic wind model derived by Chevalier and Clegg in (CC85) assumes 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^−Δ within the starburst volume R, providing solutions for Δ = 0, 1/2, 1, 3/2, and 2. In marked contrast with the CC85 model (Δ = 0), which predicts zero velocity at the centre, for a singular isothermal sphere profile (Δ = 2), we find that the flow maintains a constant Mach number of throughout the volume. The fast interior flow can be written as , where v_∞ is the asymptotic velocity, and and are the total energy and mass injection rates. For , 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 Δ = 0 (CC85) and Δ = 2 models. Next generation X-ray space telescopes such as XRISM may assess these kinematic predictions.