This paper introduces a new class of analytic MHD solutions for steady, axisymmetric, rotating outflows interacting with poleward deflected, partially open magnetic fields. In the first paper of this series, several simplifying assumptions were necessary to deal with the problem in an analytical fashion. One of the most important restrictive hypotheses is the spherical symmetry of the Mach-Alfven surfaces. This condition is relaxed in this work, and general colatitude-dependent Alfvenic surfaces are assumed. As an example of the method, approximate solutions are found for a purely radial magnetic configuration and for initially superalfvenic outflows, by assuming a small-amplitude anisotropic component of the Mach-Alfven function regarding the spherically symmetric part. It is shown that the general behavior of the stellar wind drastically changes. Particularly, for the sample of solutions derived here it is found that the terminal velocity of the wind does not diverge and increases with rotation. The energy distribution needed to sustain the outflow consists of a heating source located close to the photosphere, and its rate is much smaller than the blackbody power radiated by the star. The temperature profile displays the typical chromospheric structure of early-type stars once its associated set of boundary conditions is given.
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