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The paradigm in which magnetic fields play a crucial role inlaunching/collimating outflows in many astrophysical objects continues to gainsupport. However, semi-analytical models including the effect of magneticfields on the dynamics and morphology of jets are still missing due to theintrinsic difficulties in integrating the equations describing a collimated,relativistic flow in the presence of gravity. Only few solutions have beenfound so far, due to the highly nonlinear character of the equations togetherwith the need to blindly search for singularities. These numerical problemsprevented a full exploration of the parameter space. We present a newintegration scheme to solve r-self-similar, stationary, axisymmetricmagnetohydrodynamics equations describing collimated, relativistic outflowscrossing smoothly all the singular points (Alfven point and modified slow/fastpoints). For the first time, we are able to integrate from the disk mid-planeto downstream of the modified fast point. We discuss an ensemble of jetsolutions, emphasising trends and features that can be compared to observables.We present, for the first time with a semi-analytical MHD model, solutionsshowing counter-rotation of the jet for a substantial fraction of its extent.We find diverse jet configurations with bulk Lorentz factors up to 10 andpotential sites for recollimation between $10^3-10^7$ gravitational radii. Suchextended coverage of the intervals of quantities, such as magnetic-to-thermalenergy ratios at the base or the heights/widths of the recollimation region,makes our solutions suitable for application to many different systems wherejets are launched.

A new method for extending solutions to the self-similar relativistic magnetohydrodynamics equations for black hole outflows