On three-dimensional quasi-St\"ackel Hamiltonians

A three-dimensional integrable generalization of the St\"ackel systems is proposed. The classification of such systems is obtained which results in two families. The first one is the direct sum of the two-dimensional case which is equivalent to the representation of the Schottky--Manakov top in the quasi-St\"ackel form and a trivial 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 a usual way, is a 3-dimensional generalization of the Gibbons--Tsarev system. A generalization of the quasi-St\"ackel systems to the case of any dimension is discussed.