Undecidability of the theory of Abelian groups with an automorphism

We prove undecidability of the elementary theories of:
1) (torsion-free) Abelian groups;
2) (Archimedean) ordered Abelian groups;
3) complete Abelian groups
in the signature $\langle+,\alpha(x)=y\rangle$, where + is addition and $\alpha(x)=y$ is an automorphism of the (ordered) Abelian group.