I-surfaces from surfaces with one exceptional unimodal point

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 $E_{14}$.