Free Poisson fields and their automorphisms

Let $k$ be an arbitrary field of characteristic 0. We prove that the group of automorphisms of a free Poisson field $P(x,y)$ in two variables $x,y$ over $k$ is isomorphic to the Cremona group ${\mathrm{C}\mathrm{r}}_2(k)$. We also prove that the universal enveloping algebra $P(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n{)}^e$ of a free Poisson field $P(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n)$ is a free ideal ring and give a characterization of the Poisson dependence of two elements of $P(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n)$ via universal derivatives.