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The radiatively-driven wind theory of Castor, Abbott, \& Klein (CAK theory) is now widely accepted as the mechanism producing the mass-loss from early-type (OB) stars. The fluid equation describing the solar wind, which contains a so-called ``X-type'' critical point, was first solved by Parker using an analysis of the solution topology of the differential equation describing the wind velocity; however, the CAK wind equation is a non-linear equation for the velocity gradient, so the origin and nature of the topology of the CAK critical point has been unclear. Employing a commonly used change of variables, we obtain a linear differential equation whose solution topology is easily found. We show that in fact the CAK critical point is indeed an X-type singularity like the Parker critical point. We also find that there are four previously unknown critical points (two of these are unphysical). In addition to the trans-critical solution found by CAK, which has a monotonically increasing velocity, there are sub-critical non-monotonic solutions, analogous to the Chamberlain breeze solutions for the solar wind, as well as a trans-critical monotonically decreasing solution. However, the only outflow solution that satisfies the boundary condition of zero pressure at infinite radius is the original CAK solution. Thus the new solutions are relevant only for accretion flows fed by an external source such as a mass-transfer binary.
