  We investigate the axisymmetric magnetosphere of an aligned rotating magnetic
dipole surrounded by an ideal force-free plasma. We concentrate on the magnetic
field structure around the point of intersection of the separatrix between the
open and closed field-line regions and the equatorial plane. We first study the
case where this intersection point is located at the Light Cylinder. We find
that in this case the separatrix equilibrium condition implies that all the
poloidal current must return to the pulsar in the open-field region, i.e., that
there should be no finite current carried by the separatrix/equator current
sheet. We then perform an asymptotic analysis of the pulsar equation near the
intersection point and find a unique self-similar solution; however, a Light
Surface inevitably emerges right outside the Light Cylinder. We then perform a
similar analysis for the situation where the intersection point lies somewhere
inside the Light Cylinder, in which case a finite current flowing along the
separatrix and the equator is allowed. We find a very simple behavior in this
case, characterized by a 90-degree angle between the separatrix and the equator
and by finite vertical field in the closed-field region. Finally, we discuss
the implications of our results for global numerical studies of pulsar
magnetospheres.
