Surjective derivations in small dimensions

Inspired by a result of D. Cerveau on surjective derivations on a polynomial ring in two variables over a complex number field C, we consider a surjective derivation defined on an affine domain over C of dimension one or two. Though our proofs are mostly algebraic or algebro-geometric, the idea using a result of Dimca-Saito which is behind the arguments of Cerveau and based on the differential complex of the polynomial ring in n variables is inspiring and affects our arguments.