Derivations of polynomial algebras without Darboux polynomials

We present several new examples of homogeneous derivations of a polynomial ring k[X]=k[x1,…,xn]k[X]=k[x1,…,xn] over a field kk of characteristic zero without Darboux polynomials. Using a modification of a result of Shamsuddin, we produce these examples by induction on the number nn of variables, thus more easily than the previously known example multidimensional Jouanolou systems of Żoła̧dek.