On centralizers of elements in the Lie algebra W2(K)

Let K be a field of characteristic zero and K[x, y] the polynomial ring. Denote by W2(K) the Lie algebra of all K-derivations of K[x, y]. Centralizers of elements and maximal abelian subalgebras of the algebra W2(K) are studied. The structure of a centralizer CW2(D) depends on the kerD in the field of rational functions K(x, y) (the derivation D can be naturally extended on K(x, y)). In particular, if kerD in K(x, y) coincides with K or does not contain nonconstant polynomials, then CW2(D) is of finite dimension over K.