**Andronikos Paliathanasis**

Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban 4000, South Africa; anpaliat@phys.uoa.gr

Received: 9 July 2019; Accepted: 2 September 2019; Published: 3 September 2019

**Abstract:** We derive the one-dimensional optimal system for a system of three partial differential equations, which describe the two-dimensional rotating ideal gas with polytropic parameter *γ* > 2. The Lie symmetries and the one-dimensional optimal system are determined for the nonrotating and rotating systems. We compare the results, and we find that when there is no Coriolis force, the system admits eight Lie point symmetries, while the rotating system admits seven Lie point symmetries. Consequently, the two systems are not algebraic equivalent as in the case of *γ* = 2 , which was found by previous studies. For the one-dimensional optimal system, we determine all the Lie invariants, while we demonstrate our results by reducing the system of partial differential equations into a system of first-order ordinary differential equations, which can be solved by quadratures.

**Keywords:** lie symmetries; invariants; shallow water; similarity solutions; optimal system
