  A simplified model for the stationary, axisymmetric structure of magnetized
winds with a polytropic equation of state is presented. The shape of the
magnetic surfaces is assumed to be known (conical in this paper) within the
fast magnetosonic surface. The model is non-self-similar. The regularity of the
magnetic surfaces at critical surfaces is ensured by the Alfv\`en regularity
condition and criticality conditions at the slow and fast magnetosonic critical
points. These conditions are used to evaluate three constants of motion, the
total energy, angular momentum, and the ratio of mass to magnetic flux
$\alpha$, as well as the shape of the critical surfaces. The rotation rate
$\Omega$ and the polytropic constant $Q$ as a function of the magnetic
surfaces, together with the mass-to-magnetic flux ratio on the axis $\alpha_0$
entirely specify the model. Analytic results are given for limiting cases, and
parameter studies are performed by numerical means. The model accepts any
boundary conditions. Given the properties of astrophysical objects with
outflows, the model allows their classification in terms of a rotation
parameter. Critical surfaces are nearly spherical for slow rotators, but become
strongly distorted for rapid rotators. The fast point remains at a finite
distance for finite entropy flows, in contrast to cold flows.It is found that
for a given mass loss rate, the rotation rate is limited.
