Let $p,q \in k[x,y]$, $k$ is a field of characteristic zero. By definition, $p,q$ is a Jacobian pair if their Jacobian is invertible in $k[x,y]$, namely, $p_xq_y-p_yq_x \in k^*$, and $p,q$ is an automorphism pair if $(x,y) \mapsto (p,q)$ is an automorphism of $k[x,y]$.

There is a known result (based on S. S. Abhyankar results), Corollary 10.2.21, saying that if $p,q$ is a Jacobian pair, then there exists an automorphism $g$ of $k[x,y]$ such that $g(p)=x$ (in that case clearly $p,q$ is an automorphism pair) or the Newton polygon of $g(p)$ is contained in a rectangular $\{i,j)|0 \leq i \leq a, 0 \leq j \leq b \}$, $1 \leq a \leq b$, with $(a,b)$ belonging to the support of $g(p)$.

Assume that $g(p)$ has degree $ > 1$. By Proposition 10.2.6, there exist $1 \leq \hat{a} \leq a$ and $1 \leq \hat{b} \leq b$, such that each of $(\hat{a},0)$ and $(0,\hat{b})$ belong to the support of $g(p)$.

Is it possible that both $(a,0)$ and $(0,b)$ belong to the support of $g(p)$? (in the sub-rectangular case).

See this question.

Any help is welcome!