Tame and wild automorphisms of differential polynomial algebras of rank $2$

It is proved that the tame automorphism group of a differential polynomial algebra $k\{x,y\}$ over a field $k$ of characteristic $0$ in two variables $x$, $y$ with $m$ commuting derivations $\delta_1, \ldots, \delta_m$ is a free product with amalgamation. An example of a wild automorphism of the algebra $k\{x,y\}$ in the case of $m\geq 2$ derivations is constructed.