  We consider the Lagrangian particle model introduced in [hep-th/9612017] for
zero mass but nonvanishing second central charge of the planar Galilei group.
Extended by a magnetic vortex or a Coulomb potential the model exibits
conformal symmetry. In the former case we observe an additional SO(2,1) hidden
symmetry. By either a canonical transformation with constraints or by freezing
scale and special conformal transformations at $t=0$ we reduce the
six-dimensional phase-space to the physically required four dimensions. Then we
discuss bound states (bounded solutions) in quantum dynamics (classical
mechanics). We show that the Schr\"odinger equation for the pure vortex case
may be transformed into the Morse potential problem thus providing us with an
explanation of the hidden SO(2,1) symmetry.
