A condition for finite generation of the kernel of a derivation

We consider a homogeneous derivation on a finitely generated graded normal domain over a field, and give a sufficient condition for finite generation of its kernel. As its consequence, we show some sufficient conditions for finite generation of the kernel of a derivation on a polynomial ring. In particular, we prove that the kernel of a locally nilpotent monomial derivation on a polynomial ring in four variables is finitely generated.