A cycle in an set $X=\{x_1,\dots,x_n\}$ is a structure (X,S), where $S$ is a permutation such that $S(x_i)=x_{i+1}$ for $i<n$ and $S(x_n)=x_1$.
Now consider the theory $T:=\{ \varphi: \varphi \text{ is true in all finite cycles } (X,S)\}$. What is the shape of the infinite countable models of this theory?
Of course every infinite countable model of the theory of $(\mathbb{Z},S)$ where $S$ is the succesor function is a model of $T$. Those have the form $I\times \mathbb{Z}$ where $I$ is a countable set. Are there any others?