  We consider distinguished varieties in the bidisc $\mathbb D^2$ and the
symmetrized bidisc $\mathbb{G}_2$. A commuting pair of Hilbert space operators
$(S,P)$ for which $\Gamma=\overline{\mathbb{G}}_2$ is a spectral set is called
a \textit{$\Gamma$-contraction}. A commuting pair of contractions $(T_1,T_2)$
is said to be \textit{toral} if there is a toral polynomial $p\in \mathbb
C[z_1,z_2]$ such that $p(T_1,T_2)=0$. Similarly, a $\Gamma$-contraction $(S,P)$
is called $\Gamma$-\textit{distinguished} if $(S,P)$ is annihilated by a
$\Gamma$-distinguished polynomial in $\mathbb C[z_1,z_2]$. The main results af
this article are the following: we find a necessary and sufficient condition
such that a toral pair of contractions dilates to a toral pair of isometries.
Also, we characterize all $\Gamma$-distinguished $\Gamma$-contractions that
admit $\Gamma$-distinguished $\Gamma$-isometric dilations. We determine the
distinguished boundary of a distinguished variety in $\mathbb D^2$ and $\mathbb
G_2$. Then we prove that a pair of commuting contractions $(T_1,T_2)$ (or a
$\Gamma$-contraction $(S,P)$) dilates to a toral pair of isometries (or a
$\Gamma$-distinguished $\Gamma$-isometry) if and only if there is a toral
polynomial (or $\Gamma$-distinguished polynomial) $p\in \mathbb C[z_1,z_2]$
such that $Z(p) \cap \overline{\mathbb D}^2$ (or $Z(p) \cap \overline{\mathbb
G}_2$) is a complete spectral set for $(T_1,T_2)$ (or $(S,P)$). We provide
several examples at places to show the contrasts between the theory of toral
contractions and $\Gamma$-distinguished $\Gamma$-contractions.
