On isotropy group of locally finite derivations on $\mathbb{K}[X,Y]$

Abstract

In this paper, we study the isotropy groups of locally finite derivations of the polynomial ring $\mathbb{K}[X,Y]$, using Van den Essen's classification of locally finite derivations in two variables. We compare the isotropy group of a locally finite derivation with that of its associated exponential automorphism, showing that they coincide in the locally nilpotent case, whereas they may differ when the semisimple part is nontrivial. We also prove that every nonzero locally finite derivation has a nontrivial isotropy group.