  We survey the Carath\'eodory extremal problem $\mathrm{Car} \delta$ on the
symmetrized bidisc $$ G = \{(z+w,zw):|z|<1, \, |w|<1\} = \{(s,p)\in
\mathbb{C}^2: |s-\bar s p| < 1-|p|^2\}. $$ We also give some new results on
this topic. We are particularly interested in cases of this problem in which
the solution of the problem is not unique. It is known that, for any
$\delta=(\lambda,v)\in TG$ with $v\neq 0$, there is at least one
$\omega\in\mathbb{T}$ such that $\Phi_\omega$ solves $\mathrm{Car} \delta$,
where $\Phi_\omega(s,p) = \frac{2\omega p-s}{2-\omega s}$. Moreover, there is
an essentially unique solution of $\mathrm{Car} \delta$ if and only if $\delta$
has exactly one Carath\'eodory extremal function of the form $\Phi_\omega$ for
some $\omega\in\mathbb{T}$. We give a description of Carath\'eodory extremals
for $\delta\in TG$ with more than one Carath\'eodory extremal function
$\Phi_\omega$ for some values of $\omega \in\mathbb{T}$. The proof exploits a
model formula for the Schur class of $G$ which is an analog of the well-known
network realization formula for Schur-class functions on the disc.
