  The paradigm in which magnetic fields play a crucial role in
launching/collimating outflows in many astrophysical objects continues to gain
support. However, semi-analytical models including the effect of magnetic
fields on the dynamics and morphology of jets are still missing due to the
intrinsic difficulties in integrating the equations describing a collimated,
relativistic flow in the presence of gravity. Only few solutions have been
found so far, due to the highly nonlinear character of the equations together
with the need to blindly search for singularities. These numerical problems
prevented a full exploration of the parameter space. We present a new
integration scheme to solve r-self-similar, stationary, axisymmetric
magnetohydrodynamics equations describing collimated, relativistic outflows
crossing smoothly all the singular points (Alfven point and modified slow/fast
points). For the first time, we are able to integrate from the disk mid-plane
to downstream of the modified fast point. We discuss an ensemble of jet
solutions, emphasising trends and features that can be compared to observables.
We present, for the first time with a semi-analytical MHD model, solutions
showing counter-rotation of the jet for a substantial fraction of its extent.
We find diverse jet configurations with bulk Lorentz factors up to 10 and
potential sites for recollimation between $10^3-10^7$ gravitational radii. Such
extended coverage of the intervals of quantities, such as magnetic-to-thermal
energy ratios at the base or the heights/widths of the recollimation region,
makes our solutions suitable for application to many different systems where
jets are launched.
