In this paper we complete the analysis of the stability of periodic points of piecewise isometric systems that are defined in the entire Euclidean space. We focus on the case where the determinant of the matrix ${\rm Id}-R_{w}$ is zero, where $R_{w}$ is the linear part of the return map generated by the periodic point whose coding is made of a countable repetition of the block $w$. The case where the determinant of ${\rm Id}-R_{w}$ is non-zero was studied in Mendes and Nicol (Periodicity and recurrence in piecewise rotations of Euclidean spaces. Int. J. Bif. Chaos, 14(7), 2353–2361) but still the statement is included here for the sake of completeness.