  We present numerical solutions for stationary and axisymmetric equilibriums
of compact stars associated with extremely strong magnetic fields. The interior
of the compact stars is assumed to satisfy ideal magnetohydrodynamic (MHD)
conditions, while in the region of negligible mass density the force-free
conditions or electromagnetic vacuum are assumed. Solving all components of
Einstein's equations, Maxwell's equations, ideal MHD equations, and force-free
conditions, equilibriums of rotating compact stars associated with mixed
poloidal and toroidal magnetic fields are obtained. It is found that in the
extreme cases the strong mixed magnetic fields concentrating in a toroidal
region near the equatorial surface expel the matter and form a force-free
toroidal magnetotunnel. We also introduce a new differential rotation law for
computing solutions associated with force-free magnetosphere, and present other
extreme models without the magnetotunnel.
