The Polarization Theorem, Corollary 5.5, states that a $k$-endomorphism $\theta$ of the first Weyl algebra, $A_1(k)$, where $k$ is a field of characteristic zero, is an automorphism or there exists a positive integer $m$ and a map $\psi$ such that for some $l\neq 0$, $f_{1,-1}(\psi^{(m)}(\theta(y)))=\beta yx^{1+l/m}$, $f_{1,-1}(\psi^{(m)}(\theta(x)))=-(m/\beta l) x^{-l/m}$, $\beta \in k-\{0\}$.

Is it known that a similar theorem holds for $k[x,y]$?