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][1], 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][2], 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][3].

Any help is welcome!


  [1]: https://books.google.co.il/books/about/Polynomial_Automorphisms.html?id=wKJqqd8t8KUC&redir_esc=y
  [2]: https://books.google.co.il/books?id=MjQDCAAAQBAJ&pg=PR3&lpg=PR3&dq=free%20book%20van%20den%20essen&source=bl&ots=aWppCixEMM&sig=jCTfKAGghAnc3ShM_J4yVR6Tdh0&hl=en&sa=X&ved=0ahUKEwjW2suAxsLZAhWIUlAKHfA2DvsQ6AEIWTAJ#v=onepage&q=proposition%2010.2.21&f=false
  [3]: https://math.stackexchange.com/questions/2671131/newton-polygon-of-a-jacobian-pair