  Recently the Fibonacci word $W$ on an infinite alphabet was introduced by
[Zhang et al., Electronic J. Combinatorics 24-2 (2017) #P2.52] as a fixed point
of the morphism $\phi: (2i) \mapsto (2i)(2i+ 1),\ (2i+ 1) \mapsto (2i+ 2)$ over
all $i \in \mathbb{N}$. In this paper we investigate the occurrence of squares,
palindromes, and Lyndon factors in this infinite word.
