Abstract In this paper we will show that for every cut I of any countable nonstandard model $\mathcal {M}$ of $\mathrm {I}\Sigma _{1}$ , each I-small $\Sigma _{1}$ -elementary submodel of $\mathcal {M}$ is of the form of the set of fixed points of some proper initial self-embedding of $\mathcal {M}$ iff I is a strong cut of $\mathcal {M}$ . Especially, this feature will provide us with some equivalent conditions with the strongness of the standard cut in a given countable model $\mathcal {M}$ of $ \mathrm {I}\Sigma _{1} $ . In addition, we will find some criteria for extendability of initial self-embeddings of countable nonstandard models of $ \mathrm {I}\Sigma _{1} $ to larger models.