On three-dimensional quasi-Stackel Hamiltonians

A three-dimensional integrable generalization of the Stackel systems is proposed. A classification of such systems is obtained, which results in two families. The first family is the direct sum of the two-dimensional system which is equivalent to the representation of the Schottky-Manakov top in the quasi-Stackel form and a Stackel one-dimensional system. The second family is probably a new three-dimensional system. The system of hydrodynamic type, which we get from this family in the usual way, is a three-dimensional generalization of the Gibbons-Tsarev system. A generalization of the quasi-Stackel systems to the case of any dimension is discussed.