Darboux theory of integrability for polynomial vector fields on Sn

This is a survey on the Darboux theory of integrability for polynomial vector fields, first in R-n and second in the n-dimensional sphere S-n. We also provide new results about the maximum number of invariant parallels and meridians that a polynomial vector field X on S-n can have in function of its degree. These results in some sense extend the known result on the maximum number of hyperplanes that a polynomial vector field Y in R-n can have in function of the degree of Y.