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We extend our previous study of equilibrium solutions of non-rotatingforce-free magnetospheres of neutron stars. We show that multiple solutionsexist for the same sets of parameters, implying that the solutions aredegenerate. We are able to obtain configurations with disconnected field lines,however, in nearly all cases these correspond to degenerate higher energysolutions. We carry out a wide parametric search in order to understand theproperties of the solutions. We confirm our previous results that the lowerenergy solutions have up to $\sim 25\%$ more energy than the vacuum case,helicity of the order of $\sim 5$ (in some defined units), maximum twist of$\sim 1.5$ rad, and a dipole strength that is up to $\sim 40\%$ larger than thevacuum dipole. Including the degenerate higher energy solutions allows forlarger theoretical limits of up to $\sim 80\%$ more energy with respect to thevacuum case, helicity of the order of $\sim 8$, and a dipole strength that cannow be up to four times that of the vacuum dipole, while the twist can besignificantly larger and even diverge for configurations with disconnecteddomains. The higher energy solutions are probably unstable, therefore, it isunlikely that such magnetospheres exist under normal conditions in magnetarsand high magnetic field pulsars.

The force-free twisted magnetosphere of a neutron star - II: Degeneracies of the Grad-Shafranov equation