Darboux theory of integrability for real polynomial vector fields on the $n-$dimensional ellipsoid

We extend to the $n$-dimensional ellipsoid contained in ${\mathbb{R}}^{n+1},$ the Darboux theory of integrability for polynomial vector fields in the $n$-dimensional sphere (Llibre et al., 2018). New results on the maximum number of invariant parallels and meridians of polynomial vector fields $\text{X}$ on the invariant $n-$dimensional ellipsoid, as a function of its degree, are provided. Our results extend the known result on the upper bound for the number of invariant hyperplanes that a polynomial vector field $\text{Y}$ in ${\mathbb{R}}^n$ can have in function of the degree of $\text{Y}$.