Fritz Gesztesy and Helge Holden Real-Valued Algebro-Geometric Solutions of the Camassa--Holm hierarchy (85K, LaTeX) ABSTRACT. We provide a treatment of real-valued, smooth, and bounded algebro-geometric solutions of the Camassa--Holm (CH) hierarchy and describe the associated isospectral torus. We also discuss real-valued algebro-geometric solutions with a cusp behavior.