  The Glukhovsky-Dolzhansky (GD) model arises naturally from geophysical
science, which describes rotating fluid convection inside the ellipsoid. This
work aims to provide some new insights into the GD model. (\emph{i}) We first
show that, under some conditions there are homothetic transformations which
covert the GD model into other similar quadric physical models, therefore, our
results on the GD model can be naturally applied to the investigation of these
models. (\emph{ii}) We propose a complete classification of Darboux polynomials
and exponent factors for the GD model, which implies that the GD model has no
polynomial, rational, or Darboux first integrals. In addition, some integrable
cases of the GD model are also given when the physical parameters are allowed
to be non-positive. (\emph{iii}) The existence of global attractor is proved.
The stability and local bifurcations of all co-dimension one and two are
investigated. Particularly, we show that the GD model undergoes two dynamical
transitions as the Rayleigh number increases. (\emph{iv}) To understand the
asymptotic behavior of the orbits for the GD model, we use the Poincar\'{e}
compactification method to study its dynamical behavior at infinity. More
precisely, we prove that the phase portraits of the GD model at infinity
consist of an infinite sequence of periodic solutions and two heteroclinic
loops. Our results may help us better understand the complex and rich dynamics
of rotating fluid convection.

Dynamical Systems;Classical Analysis and ODEs