Centralizers of elements in Lie algebras of derivations of fields

Abstract

Let $K$ be an algebraically closed field of characteristic zero and $R$ an algebraic extension of the field of rational functions $K(x_1,.,x_n)$ in $n$ variables. We study centralizers of elements in the Lie algebra $Der_{K}R$ of all $K$-derivations of the field $R.$
If $F$ is the field of constants of a derivation $D\in Der _{K}R$ on $R$ and $tr.deg_{K}F\leq2$ then a characterization of the centralizer $C_{Der_{K}(R)}(D)$ is given depedent on $tr.deg_{K}F.$