Parametric CR-umbilical Locus of Ellipsoids in ${\mathbb{C}}^2$

For every real numbers $a⩾1$, $b⩾1$ with $(a,b) \neq (1,1)$,thecurveparametrizedby$\theta \in $\mathbb{R}$$valuedin$$\mathbb{C}$^2 \cong $\mathbb{R}$^4$ $$\gamma \text{\hspace{0.17em}⁣}:\theta ⟼(x(\theta )+\sqrt{-1}y(\theta ),u(\theta )+\sqrt{-1}v(\theta ))$$ with components: $$x(\theta ):=\sqrt{\frac{a-1}{a(ab-1)}}\cos \theta ,y(\theta ):=\sqrt{\frac{b(a-1)}{ab-1}}\sin \theta ,u(\theta ):=\sqrt{\frac{b-1}{b(ab-1)}}\sin \theta ,v(\theta ):=-\sqrt{\frac{a(b-1)}{ab-1}}\cos \theta ,$$ has image contained in the CR-umbilical locus: $$\gamma (\mathbb{R})\subset \mathsf{U}\mathsf{m}\mathsf{b}\mathsf{C}\mathsf{R}\left({\mathsf{E}}_{a,b}\right)\subset {\mathsf{E}}_{a,b}$$ of the ellipsoid ${\mathsf{E}}_{a,b}\subset {\mathbb{C}}^2$ of equation $a{x}^2+y^2+b{u}^2+y^2=1$.