  An exact model for magnetized and rotating outflows, underpressured at their
axis, is analysed by means of a nonlinear separation of the variables in the
two-dimensional governing magnetohydrodynamic (MHD) equations for axisymmetric
plasmas. The outflow starts subsonically and subAlfv\'enically from the central
gravitating source and its surrounding accretion disk and after crossing the
MHD critical points, high values of the Alfv\'en Mach number may be reached.
Three broad types of solutions are found: (a) collimated jet-type outflows from
efficient magnetic rotators where the outflow is confined by the magnetic hoop
stress; (b) collimated outflows from inefficient magnetic rotators where the
outflow is cylindrically confined by thermal pressure gradients; and (c)
radially expanding wind-type outflows analogous to the solar wind. In most of
the cases examined cylindrically collimated (jet-type) outflows are naturally
emerging with thermal and magnetic effects competing in the acceleration and
the confinement of the jet. The interplay of all MHD volumetric forces in
accelerating and confining the jet is displayed along all its length and for
several parameters. The solutions may be used for a physical understanding of
astrophysical outflows, such as those associated with young stellar objects,
planetary nebulae, extragalactic jets, etc.
