On gapped repeats in a cyclic Fibonacci word

In this article, we consider the words with cyclic indices. For given $s$, we consider the pair $(\iota,\kappa)$ of indices such that the word of length $s$ from $\iota$ is equal to the word of length $s$ from $\kappa$. We give a characterization of such pairs for a cyclic Fibonacci word, and give the number of them.