A subclass of solutions for equations of a reduced atmospheric model

We deal with a special subclass of solutions for the three-dimensional system of ideal polytropic gas equations corresponding to an atmospheric model. The properties of such solutions are completely characterized by a nonlinear system of ordinary differential equations of higher order. We establish that, in contrast to the corresponding two-dimensional model, all its singular points are unstable. We also find some first integrals of this system. It is shown that, in the case of axial symmetry, it could be reduced to a single equation. The system is integrable in the particular case when the adiabatic exponent is equal to 2.