Darboux theory of integrability for real polynomial vector fields on Sⁿ

Abstract

This is a survey on the Darboux theory of integrability for polynomial vector fields, first in Rⁿ and second in the n-dimensional sphere Sⁿ. We also provide new results about the maximum number of invariant parallels and meridians that a polynomial vector field X on Sⁿ 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 Rⁿ can have in function of the degree of Y.