Poisson algebras, Weyl algebras and Jacobi pairs

We study Jacobi pairs in details and obtained some properties. We also study the natural Poisson algebra structure $(\PP,[...,...],...)$ on the space $\PP:=\C[y]((x^{-\frac1N}))$ for some sufficient large $N$, and introduce some automorphisms of $(\PP,[...,...],...)$ which are (possibly infinite but well-defined) products of the automorphisms of forms $e^{\ad_H}$ for $H\in x^{1-\frac1N}\C[y][[x^{-\frac1N}]]$ and $\tau_c:(x,y)\mapsto(x,y-cx^{-1})$ for some $c\in\C$. These automorphisms are used as tools to study Jacobi pairs in $\PP$. In particular, starting from a Jacobi pair $(F,G)$ in $\C[x,y]$ which violates the two-dimensional Jacobian conjecture, by applying some variable change $(x,y)\mapsto\big(x^{b},x^{1-b}(y+a_1 x^{-b_1}+...+a_kx^{-b_k})\big)$ for some $b,b_i\in\Q_+,a_i\in\C$ with $b_i<1<b$, we obtain a \QJ pair still denoted by $(F,G)$ in $\C[x^{\pm\frac1N},y]$ with the form $F=x^{\frac{m}{m+n}}(f+F_0)$, $G=x^{\frac{n}{m+n}}(g+G_0)$ for some positive integers $m,n$, and $f,g\in\C[y]$, $F_0,G_0\in x^{-\frac1N}\C[x^{-\frac1N},y]$, such that $F,G$ satisfy some additional conditions. Then we generalize the results to the Weyl algebra $\WW=\C[v]((u^{-\frac1N}))$ with relation $[u,v]=1$, and obtain some properties of pairs $(F,G)$ satisfying $[F,G]=1$, referred to as Dixmier pairs.