Integrability analysis of a simple model for describing convection of a rotating fluid

We study the Darboux integrability of a simple system of three ordinary differential equations called the Glukhovsky-Dolzhansky system, which describes a three-mode model of rotating fluid convection inside the ellipsoid. (1) Our results show that it has no polynomial, rational, or Darboux first integrals for any value of parameters in the physical sense, that is, positive parameters. (2) We also provide some integrable cases of this model when parameters are allowed to be non-positive. (3) We finally give some links between the Glukhovsky-Dolzhansky system and other similar systems in $\mathbb{R}^3$, which admits rotational symmetry and has three nonlinear cross terms