Darboux theory of integrability for polynomial vector fields on 𝕊n

Abstract

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