  We resume the study initiated in \cite{CL}. For a generic curve $C$ in an
ample linear system $\vert \mathcal{L} \vert$ on a toric surface $X$, a
vanishing cycle of $C$ is an isotopy class of simple closed curve that can be
contracted to a point along a degeneration of $C$ to a nodal curve in $\vert
\mathcal{L} \vert$. The obstructions that prevent a simple closed curve in $C$
from being a vanishing cycle are encoded by the adjoint line bundle $K_X
\otimes \mathcal{L}$. In this paper, we consider the linear systems carrying
the two simplest types of obstruction. Geometrically, these obstructions
manifest on $C$ respectively as an hyperelliptic involution and as a Spin
structure. In both cases, we determine all the vanishing cycles by
investigating the associated monodromy maps, whose target space is the mapping
class group $MCG(C)$. We show that the image of the monodromy is the subgroup
of $MCG(C)$ preserving respectively the hyperelliptic involution and the Spin
structure. In particular, we provide an explicit finite set of generators for
the Spin mapping class group. The results obtained here support the Conjecture
$1$ in \cite{CL} aiming to describe all the vanishing cycles for any pair $(X,
\mathcal{L})$.
