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

Equinormalizable theory for plane curve singularities with embedded points and the theory of equisingularity