We can vary a position of $(x_{0},0)$, thus obtaining a continuous path in the space of ${\bf Z}$-affine structures with singularities in ${{\bf R}}^{2}$.

More generally, suppose that $B$ is equipped with a ${\bf Z}$-affine structure which has standard singularities at points $p_{1},\dots,p_{m}$. Then we can slightly move each point $p_{i}$ in the direction invariant under the local monodromy around $p_{i}$. This gives a new ${\bf Z}$-affine structure which is ZPL-isomorphic to the initial one.