  We introduce a class of complex surface singularities - the blow-$ADE$
singularities - which are likely to be stable with respect to $\mu^*$-constant
deformations. We prove such a stability property in several special cases.
Here, we emphasize that we are not just considering deformation families for
small values of the deformation parameter but families connecting any two
elements in the $\mu^*$-constant stratum.
