  We complement recent work of Gallardo, Pearlstein, Schaffler, and Zhang,
showing that the stable surfaces with $K_X^2 =1$ and $\chi(\mathcal O_X) = 3$
they construct are indeed the only ones arising from imposing an exceptional
unimodal double point.
  In addition, we explicitly describe the birational type of the surfaces
constructed from singularities of type $E_{12}$, $E_{13}$, and $E_{14}$.

Algebraic Geometry