Model companion of ordered theories with an automorphism

Kikyo and Shelah showed that if $T$ is a theory with the Strict Order Property in some first-order language $\mathcal {L}$, then in the expanded language $\mathcal {L}_\sigma := \mathcal {L}\cup \{\sigma \}$ with a new unary function symbol $\sigma$, the bigger theory $T_\sigma := T\cup \{\lq\lq \sigma$$\mbox { is an } \mathcal {L}\mbox {-automorphism''}\}$ does not have a model companion. We show in this paper that if, however, we restrict the automorphism and consider the theory $T_\sigma$ as the base theory $T$ together with a ``restricted'' class of automorphisms, then $T_\sigma$ can have a model companion in $\mathcal {L}_\sigma$. We show this in the context of linear orders and ordered abelian groups.