  Hayut and first author isolated the notion of a critical cardinal in [1]. In
this work we answer several questions raised in the original paper. We show
that it is consistent for a critical cardinals to not have any ultrapower
elementary embeddings, as well as that it is consistent that no target model is
closed. We also prove that if $\kappa$ is a critical point by any ultrapower
embedding, then it is the critical point by a normal ultrapower embedding. The
paper contains several open questions of interest in the study of critical
cardinals.
