  Let $\delta$ be a derivation in a $K$-algebra $R$ and let ${\rm
Aut_{\delta}}(R)$ be the isotropy group with respect to the natural conjugation
action of $\rm Aut(R)$ of $K$-automorphisms on the set $\rm Der(R)$ of
$K$-derivations: that is, the subgroup of automorphisms that commute with the
derivation. We explore the characterization of ${\rm Aut_{\delta}}(R)$ for
quantum Weyl algebras and we prove that in the case of the Jordanian plane it
is always a finite cyclic group. Furthermore, we obtain an arithmetic
characterization that allows us to trivially demonstrate that these algebras
are not isomorphic.
