The Polarization Theorem, [Corollary 5.5][1], states that a $k$-endomorphism of the first Weyl algebra, $A_1(k)$, where $k$ is a field of characteristic zero, is an automorphism or has some special property.

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


  [1]: https://www.jstor.org/stable/2373768?seq=1#page_scan_tab_contents