Non-Homogeneous Hydrodynamic Systems and Quasi-St\"ackel Hamiltonians

In this paper we present a novel construction of non-homogeneous hydrodynamic equations from what we call quasi-St\"ackel systems, that is non-commutatively integrable systems constructed from appropriate maximally superintegrable St\"ackel systems. We describe the relations between Poisson algebras generated by quasi-St\"ackel Hamiltonians and the corresponding Lie algebras of vector fields of non-homogeneous hydrodynamic systems. We also apply St\"ackel transform to obtain new non-homogeneous equations of considered type.