  The method proposed by Inomata and his collaborators allows us to transform a
damped Caldiroli-Kanai oscillator with time-dependent frequency to one with
constant frequency and no friction by redefining the time variable, obtained by
solving a Ermakov-Milne-Pinney equation. Their mapping ``Eisenhart-Duval''
lifts as a conformal transformation between two appropriate Bargmann spaces.
The quantum propagator is calculated also by bringing the quadratic system to
free form by another time-dependent Bargmann-conformal transformation which
generalizes the one introduced before by Niederer and is related to the mapping
proposed by Arnold. Our approach allows us to extend the Maslov phase
correction to arbitrary time-dependent frequency. The method is illustrated by
the Mathieu profile.
