  We identify a strong similarity among several distinct originally
second-class systems, including both mechanical and field theory models, which
can be naturally described in a gauge-invariant way. The canonical structure of
such related systems is encoded into a gauge-invariant generalization of the
quantum rigid rotor. We pursue the BRST symmetry analysis and the BFV
functional quantization for the mentioned gauge-invariant version of the
generalized quantum rigid rotor. We obtain different equivalent effective
actions according to specific gauge-fixing choices, showing explicitly their
BRST symmetries. We apply and exemplify the ideas discussed to two particular
models, namely, motion along an elliptical path and the $O(N)$ nonlinear sigma
model, showing that our results reproduce and connect previously unrelated
known gauge-invariant systems.
