Let $k$ be a field and $(A,m)$ be the completion of the local ring of a smooth point of a $k$-variety. Let $x_1,x_2\in m\backslash m^2$ be regular elements. I am interested in knowing if one can find a $k$-linear automorphism of $A$ which takes the ideal $(x_1)$ to $(x_2)$.

If $k$ is perfect, it is easy to see that this can be done, since there is a $k$-linear embedding $A/m\hookrightarrow A$ which induces a $k$-linear isomorphism $$ A \cong (A/m)[|x_1,...,x_n|] $$ for any choice of regular sequence $(x_1,..,x_n)$ in $A$.

When $k$ is not perfect, I can hardly find any non-trivial $k$-automorphism of $(A,m)$.