  Starting with a solvable Nevanlinna-Pick interpolation problem with the
initial data coming from the symmetrized bidisk, this paper studies the
corresponding uniqueness set, i.e., the largest set in the domain where all
solutions to the problem coincide. It is shown that the uniqueness set
coincides with an algebraic variety in the domain. The algebraic variety -
canonically constructed from the interpolation data - is called the uniqueness
variety. It was shown that the uniqueness variety contains a distinguished
variety which by definition is the zero set of a two-variable polynomial that
intersects the domain and exits through its distinguished boundary. A complete
algebraic and geometric characterizations of distinguished varieties are
obtained in this paper.
