Infinite transitivity and polynomial vector fields

We prove that for many pairs $H_1,H_2$ of root subgroups of the automorphism group $\text{Aut}({\mathbb{C}}^2)$ the diagonal action of the group generated by $H_1,H_2$ on $({\mathbb{C}}^2{)}^m$ has an open orbit for any positive integer $m$. The result is based on the study of the Lie algebra of polynomials in two variables with the standard Poisson bracket.