  An I-surface $S$ is an algebraic surface of general type with $K_S^2 = 1$ and
$p_g(S) = 2$. Recent research has centered on trying to give an explicit
description of the KSBA compactification of the moduli space of these surfaces.
The possible normal Gorenstein examples have been enumerated by work of
Franciosi-Pardini-Rollenske. The goal of this paper is to give a more precise
description of such surfaces in case their singularities are simple elliptic
and/or cusp singularities, and to work out their deformation theory. In
particular, under some mild general position assumptions, we show that
deformations of the surfaces in question are versal for deformations of the
singular points, with two exceptions where the discrepancy is analyzed in
detail.
