  Motivated by the recent progress of research on extending holomorphic
functions defined on subvarieties of classical domains and its connections to
the 3-point Pick interpolation, we study a special class of two-dimensional
algebraic subvarieties $M_\alpha$ of the unit tridisc, defined as the sets
$$\lbrace (z_1,z_2,z_3)\in
\mathbb{D}^3:\alpha_1z_1+\alpha_2z_2+\alpha_3z_3=\overline{\alpha}_1z_2z_3+\overline{\alpha}_2z_1z_3+\overline{\alpha}_3z_1z_2\rbrace.$$
In this paper we show that given non-degenerated extremal maximal $3$-point
Pick problem there exists an $\alpha$ such that $M_\alpha$ appears as its
uniqueness variety. We also describe several geometric properties of $M_\alpha$
and show the biholomorphic equivalence between any two surfaces $M_\alpha$ and
$M_\beta$, where the triples $\alpha$ and $\beta$ satisfy the so called
triangle inequality.

Complex Variables