  The minimal Hamiltonian for a family of relativistic rotators is constructed
by a direct application of the Dirac procedure for constrained systems. The
Hamiltonian equations can be easily solved. It is found that the resulting
motion is unique and qualitatively the same for all phenomenological rotators,
only the relation between mass and spin is different. There is a critical point
in the construction when such a relation cannot be established, implying that
the number of primary constraints is greater. In that case the mass and the
spin become unrelated, separately fixed parameters, and the corresponding
Hamiltonian changes qualitatively. Furthermore, a genuine physical observable
becomes a gauge variable. This paradoxical result is consistent with the fact
already known at the Lagrangian level that the Hessian rank is lower than
expected, and the equations of motion indeterminate on
$\mathbb{R}^3\times\mathbb{S}^2$.
