  We consider a second degree algebraic curve describing a general conic
constraint imposed on the motion of a massive spinless particle. The problem is
trivial at classical level but becomes involved and interesting in its quantum
counterpart with subtleties in its symplectic structure and symmetries. The
model is used here to investigate quantization issues related to the
Hamiltonian constraint structure, Dirac brackets, gauge symmetry and BRST
transformations. We pursue the complete constraint analysis in phase space and
perform the Faddeev-Jackiw symplectic quantization following the
Barcelos-Wotzasek iteration program to unravel the fine tuned and more relevant
aspects of the constraint structure. A comparison with the longer usual
Dirac-Bergmann algorithm, still more well established in the literature, is
also presented. While in the standard DB approach there are four second class
constraints, in the FJBW they reduce to two. By using the symplectic potential
obtained in the last step of the FJBW iteration process we construct a gauge
invariant model exhibiting explicitly its BRST symmetry. Our results reproduce
and neatly generalize the known BRST symmetry of the rigid rotor showing that
it constitutes a particular case of a broader class of theories.
