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The ladder representations of Askey-Wilson algebra (AWA) is investigated and the overlap problem is discussed. The full class of exactly solvable potentials for one-dimensional ordinary Schrödinger equation with the AWA as an algebra of dynamical symmetry is found for creation-annihilation operators of third order. The generating spectrum algebra for lattice Schrödinger equation as AWA is examined. All exactly solvable potentials with this dynamical symmetry are found. Some generalizations of obtained results are discussed.