  We consider the class of spinning particle theories, whose quantization
corresponds to the continuous helicity representation of the Poincare group.
The classical trajectories of the particle are shown to lie on the parabolic
cylinder with a lightlike axis irrespectively to any specifics of the model.
The space-time position of the cylinder is determined by the values of momentum
and total angular momentum. The value of helicity determines the focal distance
of parabolic cylinder. Assuming that all the world lines lying on one and the
same cylinder are connected by gauge transformations, we derive the geometrical
equations of motion for the particle. The timelike world paths are shown to be
solutions to a single relation involving the invariants of trajectory up to
fourth order in derivatives. Geometrical equation of motion is non-Lagragian,
but it admits equivalent variational principle in the extended set of dynamical
variables. The lightlike paths are also admissible on the cylinder, but they do
not represent the classical trajectories of this spinning particle. The
classical trajectories of massless particle (with zero helicity) are shown to
lie on hyperplanes, whose spacetime position depends on momentum and total
angular momentum.
