  We prove that $\W_3$ is the gauge symmetry of the scale-invariant rigid
particle, whose action is given by the integrated extrinsic curvature of its
world line. This is achieved by showing that its equations of motion can be
written in terms of the Boussinesq operator. The $\W_3$ generators $T$ and $W$
then appear respectively as functions of the induced world line metric and the
extrinsic curvature. We also show how the equations of motion for the standard
relativistic particle arise from those of the rigid particle whenever it is
consistent to impose the ``zero-curvature gauge'', and how to rewrite them in
terms of the $\KdV$ operator. The relation between particle models and
integrable systems is further pursued in the case of the spinning particle,
whose equations of motion are closely related to the $\SKdV$ operator. We also
partially extend our analysis in the supersymmetric domain to the scale
invariant rigid particle by explicitly constructing a supercovariant version of
its action. Comment: This is an expanded version of hep-th/9406072 (to be
published in the Proceedings of the Workshop on the Geometry of Constrained
Dynamical Systems, held at the Isaac Newton Institute for Mathematical
Sciences, Cambridge, June 14-18, 1994.).
