Infinite families of 2-isometric and not 3-isometric binary words

Let Fk be the family of the binary words containing the letter 0 exactly k times. Ilić, Klavžar and Rho constructed an infinite subfamily of 2-isometric and not 3-isometric words in F2. Wei and Zhang further found all such words in F2. In this paper we find that there exists no 2-isometric and not 3-isometric word in F3. For k≠1,3,4 and 7, we also construct an infinite subfamily of 2-isometric and not 3-isometric words in Fk. Based on those results and computer experiments, we conjecture that F1, F3, F4 and F7 are the only families in which there exists no 2-isometric and not 3-isometric word.