After asking this question, I would like to generalize it a little:
Let $k$ be a field of characteristic zero. A $k$-automorphism of $k[[x,y]]$ was described here: $f: (x,y) \mapsto (A,B)$ is a $k$-automorphism of $k[[x,y]]$ if and only if $A,B$ are power series with no constant term whose linear parts are linearly independent.
When a given $k$-automorphism $f:(x,y) \mapsto (A,B)$ of $k[[x,y]]$ is also an automorphism of $k[x,y]$?
More precisely, what additional conditions are needed in order to answer my question positively?
My question does not have a uniqe answer; for example: (1) In the above mentioned question we have seen that if $f$ is of finite order and $A,B \in k[x,y]$, then $f$ is also an automorphism of $k[x,y]$. (2) I wonder if the following condition is sufficient: $f$ is of finite order (apriori, we do not know if $A,B \in k[x,y]$) and there exists $T \subseteq k[x,y]$ such that $T$ is dense in $k[[x,y]]$ and invariant under $f$, namely $f(T) \subseteq T$. Notice that if $T=k[x,y]$, then we arrive at the first example (1), so we better assume that $T \subsetneq k[x,y]$ if we wish to get a new condition.
Thank you very much!
