  We construct a symplectic, globally defined, minimal-coordinate, equivariant
integrator on products of 2-spheres. Examples of corresponding Hamiltonian
systems, called spin systems, include the reduced free rigid body, the motion
of point vortices on a sphere, and the classical Heisenberg spin chain, a
spatial discretisation of the Landau-Lifschitz equation. The existence of such
an integrator is remarkable, as the sphere is neither a vector space, nor a
cotangent bundle, has no global coordinate chart, and its symplectic form is
not even exact. Moreover, the formulation of the integrator is very simple, and
resembles the geodesic midpoint method, although the latter is not symplectic.
