  This paper concerns the CR umbilical locus of a real ellipsoid in
$\mathbb{C}^2$, the set of points at which the ellipsoid can be osculated by a
biholomorphic image of the sphere up to 6th order. Huang and Ji proved that
this locus is non-empty. Ebenfelt and Zaitsev proved that, for ellipsoids that
are sufficiently "closed" to the sphere, the locus actually contains a "stable"
curve of umbilical points. Foo, Merker and Ta later provided an explicit curve
that is contained in it. The main result in this paper exhibits that the
umbilical locus is the union of the curves mentioned above and a non-trivial
real variety which is defined by two homogeneous real sextic equations of four
real variables. When there are suitable pairs of equal semi-axes, one of them
factors allowing us to determine the locus explicitly.
