A Subclass of Solutions for Equations of a Reduced Atmospheric Model

A special subclass of solutions of the three-dimensional system of ideal polytropic gas equations corresponding to an atmospheric model is considered. The properties of these solutions are completely characterized by a high-order nonlinear system of ordinary differential equations. Unlike the corresponding two-dimensional model, all singular points of this system have been found to be unstable. Some first integrals of this system have been found. In the case of axial symmetry, the system can be reduced to a single equation. If the adiabatic exponent is equal to 2, the system is integrable.