Correction to Generic polynomial vector fields are not integrable

In the paper Generic polynomial vector fields are not integrable [1], we study some generic aspects of polynomial vector fields or polynomial derivations with respect to their integration. Using direct sums of derivations together with our previous results we showed that, for all n ≥ 3 and s ≥ 2, the absence of polynomial first integrals, or even of Darboux polynomials, is generic for homogeneous polynomial vector fields of degree s in n variables. To achieve this task, we need an example of such vector fields of degree s ≥ 2 for any prime number n ≥ 3 of variables and also for n = 4. The purpose of this note is to correct a gap in our paper for n = 4 by completing the corresponding proof.