  The Author shows how to construct a class of Lagrangians for relativistic
dynamical systems described by position and a single spinor. One arrives to it
by imposing three requirements: 1) Hamilton action should be reparametrization
invariant, 2) the number of dimensional parameters should be minimal, 3) the
spinor phase should be a cyclic variable. In more detail are discussed those of
the the Lagrangians which depend on the spinor's null vector and its worldline.
An interesting relation between a Hessian determinant and Casimir invariants
for such objects leads to the conclusion that no fundamental objects of this
kind exist with worldlines uniquely determinable from the Hamilton action and
the initial conditions. This unexpected result poses the general question about
existence of classical fundamental dynamical systems with well posed Cauchy
problem. [A relativistic dynamical system is said to be fundamental if its
Casimir invariants are parameters, not constants of motion]
