[en] The object of our study is the set of Z-words, that is, (bi)infinite sequences of alphabetic symbols indexed by integers. We consider an ordered family of subsets of the set of all the cube-free Z-words in a two-letter alphabet. The construction of this family is based on the notion of the local exponent of a Z-word. The problem of existence of cube-free Z-words which are Z-words of local exponent 2 (the minimum possible) is described. An important distinction is drawn between strongly cube-free Z-words and Z-words of greater local exponent