Chains of strongly non-reflexive dual groups of integer-valued continuous functions

Answering a question of Eklof-Mekler (Almost free modules, set-theoretic methods, North-Holland, Amsterdam, 1990), we prove: (1) If there exists a non-reflecting stationary set of $\omega _{i}$ consisting of ordinals of cofinality $\omega$ for each $1 < i < \omega$, then there exist abelian groups $A_{n} (n \in {\mathbb{Z}})$ such that $A_{n}^{\ast } \cong A_{n+1}$ and $A_{n} \not \cong A_{n+2}$ for each $n \in {\mathbb{Z}}$. (2) There exist abelian groups $A_{n} (n \in {\mathbb{Z}})$ such that $A_{n}^{\ast } \cong A_{n+1}$ for each $n \in {\mathbb{Z}}$ and $A_{n} \not \cong A_{n+2}$ for each $n < 0$. The groups $A_{n}$ are the groups of $\mathbb{Z}$-valued continuous functions on a topological space and their dual groups.