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

## TL;DR

This paper introduces a new method to construct non-homogeneous hydrodynamic equations using quasi-St"ackel systems, linking integrable Hamiltonian systems with hydrodynamics through algebraic structures.

## Contribution

It presents a novel construction of non-homogeneous hydrodynamic equations from quasi-St"ackel systems, expanding the understanding of their algebraic and geometric properties.

## Key findings

- Established relations between Poisson algebras and Lie algebras of hydrodynamic systems.
- Applied St"ackel transform to generate new non-homogeneous equations.
- Connected integrable Hamiltonian systems with hydrodynamic equations.

## Abstract

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.