  We study the equilibrium measure on the two dimensional sphere in the
presence of an external field generated by r+1 equal point charges that are
symmetrically located around the north pole. The support of the equilibrium
measure is known as the droplet. The droplet has a motherbody which we
characterize by means of a vector equilibrium problem (VEP) for r measures in
the complex plane.
  The model undergoes two transitions which is reflected in the support of the
first component of the minimizer of the VEP, namely the support can be a finite
interval containing 0, the union of two intervals, or the full half-line. The
two interval case corresponds to a droplet with two disjoint components, and it
is analyzed by means of a genus one Riemann surface.
