  In this paper we give some criteria for a family of generically reduced plane
curve singularities to be equinormalizable. The first criterion is based on the
$\delta$-invariant of a (non-reduced) curve singularity which is introduced by
Br\"{u}cker-Greuel (\cite{BG}). The second criterion is based on the
I-equisingularity of a $k$-parametric family ($k\geq 1$) of generically reduced
plane curve singularities, which is introduced by Nobile (\cite{No}) for
one-parametric families. The equivalence of some kinds of equisingularities of
a family of generically reduced plane curve singularities is also studied.
