On weakly semisimple derivations of the polynomial ring in two variables

Let $\mathbb K$ be an algebraically closed field of characteristic zero and $\mathbb K[x,y]$ the polynomial ring. Every element $f\in \mathbb K[x,y]$ determines the Jacobian derivation $D_f$ of $\mathbb K[x,y]$ by the rule $D_f(h) = det J(f,h)$, where $J(f,h)$ is the Jacobian matrix of the polynomials $f$ and $h$. A polynomial $f$ is called weakly semisimple if there exists a polynomial $g$ such that $D_f(g) = \lambda g$ for some nonzero $\lambda\in \mathbb K$. Ten years ago, Y. Stein posed a problem of describing all weakly semisimple polynomials (such a description would characterize all two dimensional nonabelian subalgebras of the Lie algebra of all derivations of $\mathbb K[x,y]$ with zero divergence). We give such a description for polynomials $f$ with the separated variables, i.e. which are of the form: $f(x,y) = f_1(x) f_2(y)$ for some $f_{1}(t), f_{2}(t)\in \mathbb K[t]$.