On plane polynomial automorphisms commuting with simple derivations

We consider the subgroup Aut(D) consisting of automorphisms of K[x,y] commuting with a derivation D, where K is an algebraically closed field of characteristic 0. We prove that if D is simple (i.e. D does not stabilize non-trivial ideals), then Aut(D)=1. In the case where D is of Shamsuddin type, this result was proven by R. Baltazar in 2014. Moreover, we show that general Shamsuddin type derivations D with Aut(D)=1 are simple.