  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.
