There is an interesting theorem about the first Weyl algebra $A_1(k)= k \langle x,y | yx-xy= 1 \rangle$, $k$ is a field of characteristic zero, the Polarization Theorem, Corollary 5.5 by A. Joseph.

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

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