  The ability of a plasma surrounding spinning black holes to extract
rotational energy and power energetic emissions has been recognized as a key
astrophysical phenomenon. Important insights into the nature of this process
are obtained through the analysis of the interplay between a force-free
magnetosphere and the black hole. This task involves solving a complicated
system of equations, often requiring complex numerical simulations. Recent
analytical attempts at tackling this problem have exploited the fact that the
near horizon region of extreme Kerr (NHEK) is endowed with an enhanced symmetry
group. We continue in this direction and show that for some conformally
self-similar solutions, the NHEK force-free equations reduce to a single
non-linear ordinary differential equation which is difficult to solve with
straightforward integration. We here introduce a new approach specifically
tailored to this problem and describe how one can obtain physically meaningful
solutions.
