2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/90975In this paper we consider $Z_3$-graded topological symmetries (TSs) in one dimensional quantum mechanics. We give a classification of one dimensional quantum systems possessing these symmetries and show that different classes correspond to a positive integer $N$.9 pages. To appear in Int. J. Mod. Phys. A. Page style and minor typos correctedQuantum Physicstext