  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 $\mathrm{Cr}_2(k)$. We also prove that the
universal enveloping algebra $P(x_1,...,x_n)^e$ of a free Poisson field
$P(x_1,...,x_n)$ is a free ideal ring and give a characterization of the
Poisson dependence of two elements of $P(x_1,...,x_n)$ via universal
derivatives.
