Darboux theory of integrability for real polynomial vector fields on ⁿ

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