  We investigate topological properties of a completely integrable system on
$S^2\times S^2 \times S^2$ which was recently shown to have a Lagrangian fiber
diffeomorphic to $\mathbb{R} P^3$ not displaceable by a Hamiltonian isotopy
[Oakley J., Ph.D. Thesis, University of Georgia, 2014]. This system can be
viewed as integrating the determinant, or alternatively, as integrating a
classical Heisenberg spin chain. We show that the system has non-trivial
topological monodromy and relate this to the geometric interpretation of its
integrals.
