  We scrutinize the possibility of extending the result of \cite{ccr} to the
case of q-deformed oscillator for $q$ real; for this we exploit the whole range
of the deformation parameter as much as possible. We split the case into two
depending on whether a solution of the commutation relation is bounded or not.
Our {\it leitmotif} is {\it subnormality}.
  The deformation parameter $q$ is reshaped and this is what makes our approach
effective. The newly arrived parameter, the operator $C$, has two remarkable
properties: it separates in the commutation relation the annihilation and
creation operators from the deformation as well as it $q$-commutes with those
two. This is why introducing the operator $C$ seems to be far-reaching.
