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Diagonal lines in symbolic recurrence plots are closely related to theidentification and characterization of specific biprolongable words within asequence. In this paper we focus on the recurrence plot of a fixed point of auniform binary substitution. We show that, if the substitution is primitive andaperiodic, the set of all diagonal line lengths of the recurrence plot has zerodensity. However, if a line of a specific length exists in the recurrence plot,the density of (the set of starting points of) all diagonal lines with thatlength is strictly positive. On the other hand, we demonstrate that therecurrence plot of a non-primitive substitution contains lines of any givenlength. Nonetheless, for any given length, the density of lines with thatlength is zero.

Symbolic recurrence plot for uniform binary substitutions