  The current classification of $\mathcal{N} = 1$ string theories in eight and
seven dimensions is completely captured by K3 surfaces with F-Theory or
M-Theory frozen singularities. In this note we show that there are inequivalent
ways of freezing certain collections of singularities which have the same ADE
type, and so there are more connected components in the moduli space than
previously thought; namely, one more in eight dimensions and three more in
seven dimensions. We argue that the new component in eight dimensions
decompactifies to a string theory in nine dimensions with rank 1 gauge group,
which has been so far unidentified. Constructing and studying the stringy
descriptions corresponding to these moduli spaces is the subject of a companion
paper.
