  For any curve $\mathcal{V}$ in a toric surface $X$, we study the critical
locus $S(\mathcal{V})$ of the moment map $\mu$ from $\mathcal{V}$ to its
compactified amoeba $\mu(\mathcal{V})$. We show that for curves $\mathcal{V}$
in a fixed complete linear system, the critical locus $S(\mathcal{V})$ is
smooth apart from some real codimension $1$ walls. We then investigate the
topological classification of pairs $(\mathcal{V},S(\mathcal{V}))$ when
$\mathcal{V}$ and $S(\mathcal{V})$ are smooth. As a main tool, we use the
Lyashko-Looijenga mapping ($\mathcal{LL}$) relative to the logarithmic Gauss
map $\gamma : \mathcal{V} \rightarrow \mathbb{C}P^1$. We prove two statements
concerning $\mathcal{LL}$ that are crucial for our study: the map
$\mathcal{LL}$ is algebraic; the map $\mathcal{LL}$ extends to nodal curves. It
allows us to construct many examples of pairs $(\mathcal{V},S(\mathcal{V}))$ by
perturbing nodal curves.
