2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/217056A set of exactly solvable one-dimensional quantum mechanical potentials is described. It is defined by a finite-difference-differential equation generating in the limiting cases the Rosen-Morse, harmonic, and Pöschl-Teller potentials. General solution includes Shabat's infinite number soliton system and leads to raising and lowering operators satisfying $q$-deformed harmonic oscillator algebra. In the latter case energy spectrum is purely exponential and physical states form a reducible representation of the quantum conformal algebra $su_q(1,1)$.8 pages, LATEX. Essentially improved versionHigh Energy Physics - Theorytext