  We classify, up to few exceptions, the orbit closures of the
$\mathrm{Mod}(\Sigma)$-action on the affine character variety
$\chi(\mathrm{Aff}(\mathbb{C}))$. We obtain from this classification that the
only obstruction for a non-abelian representation $\rho : \pi_1 \Sigma
\longrightarrow \mathrm{Aff}(\mathbb{C})$ to be the holonomy of a branched
affine structure on $\Sigma$ is to be Euclidean and not to have positive
volume, where $\Sigma$ is a closed oriented surface of genus $g \geq 2$.
