Darboux theory of integrability in \(\mathbb {T}^n\)

We develop the Darboux theory of integrability for polynomial vector fields in the n-dimensional torus \(\mathbb {T}^n\) ${\mathbb{T}}^n$ . We also provide the maximum number of invariant parallels and meridians that a polynomial vector field X on \(\mathbb {T}^n\) ${\mathbb{T}}^n$ can have in function of its degree.