  In this paper we consider the existence and uniqueness of pure states with
some fixed slope $(s, t) \in [0, 1]^2$ for a general ferroelectric six-vertex
model. First, we show there is an open subset $\mathfrak{H} \subset [0, 1]^2$,
which is parameterized by the region between two explicit hyperbolas, such that
there is no pure state for the ferroelectric six-vertex model of any slope $(s,
t) \in \mathfrak{H}$. Second, we show that there is a unique pure state for
this model of any slope $(s, t)$ on the boundary $\partial \mathfrak{H}$ of
$\mathfrak{H}$. These results confirm predictions of Bukman-Shore from 1995.
