  By means of a nonlinear separation of the variables in the governing full set
of the magnetohydrodynamic (MHD) equations for axisymmetric plasmas we analyse
an exact model for magnetized and rotating outflows which are hotter and
overpressured at their axis. These outflows start subsonically and
subAlfv\'enically from the central gravitating source and its surrounding
accretion disk. Subsequently, they accelerate thermally and
magnetocentrifugally and thus cross the appropriate MHD critical points,
reaching high values of the Alfv\'en Mach number. Three types of solutions are
found : (a) collimated jet-type outflows from efficient magnetic rotators with
the flow confined by the magnetic hoop stress; (b) radially expanding wind-type
outflows analogous to the solar wind, from inefficient magnetic rotators or
strongly overpressured sources; (c) terminated solutions with increasing
amplitude of oscillations in the width of the beam. In contrast to previously
studied underpressured outflows, the transition from collimated jets to
uncollimated winds is not continuous in the appropriate parametric space with a
gap where no stationary solution is found. Superfast at infinity solutions are
filtered by three critical surfaces corresponding to the three known limiting
characteristics or separatrices of MHD wind theory. Collimated and terminated
solutions cross the slow, Alfv\'en and fast magneto-acoustic critical points.
Radially expanding solutions cross the slow and Alfv\'en critical points while
the last boundary condition is imposed by requiring that the pressure vanishes
at infinity.
