  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.
