  There are two relativistic rotators with Casimir invariants of the
Poincar\'{e} group being fixed parameters. The particular models of spinning
particles were studied in the past both at the classical and quantum level.
Recently, a minimal interaction with electromagnetic field has been considered.
We show that the dynamical systems can be uniquely singled out from among other
relativistic rotators by the unphysical requirement that the Hessian referring
to the physical degrees of freedom should be singular. Closely related is the
fact that the equations of free motion are not independent, making the
evolution indeterminate. We show that the Hessian singularity cannot be removed
by the minimal interaction with the electromagnetic field. By making use of a
nontrivial Hessian null space, we show that a single constraint appears in the
external field for consistency of the equations of motion with the Hessian
singularity. The constraint imposes unphysical limitation on the initial
conditions and admissible motions. We discuss the mechanism of appearance of
unique solutions in external fields on an example of motion in the uniform
magnetic field. We give a simple model to illustrate that similarly constrained
evolution cannot be determinate in arbitrary fields.
