  We investigate a reciprocally invariant system proposed by Low and Govaerts
et al., whose action contains both the orthogonal and the symplectic forms and
is invariant under global $O(2,4)\cap Sp(2,4)$ transformations. We find that
the general solution to the classical equations of motion has no linear term in
the evolution parameter, $\tau$, but only the oscillatory terms, and therefore
cannot represent a particle propagating in spacetime. As a remedy, we consider
a generalisation of the action by adopting a procedure similar to that of Bars
et al., who introduced the concept of a $\tau$ derivative that is covariant
under local Sp(2) transformations between the phase space variables
$x^\mu(\tau)$ and $p^\mu (\tau)$. This system, in particular, is similar to a
rigid particle whose action contains the extrinsic curvature of the world line,
which turns out to be helical in spacetime. Another possible generalisation is
the introduction of a symplectic potential proposed by Montesinos. We show how
the latter approach is related to Kaluza-Klein theories and to the concept of
Clifford space, a manifold whose tangent space at any point is Clifford algebra
Cl(8), a promising framework for the unification of particles and forces.
