  The discrete Nahm equations, a system of matrix valued difference equations,
arose in the work of Braam and Austin on half-integral mass hyperbolic
monopoles.
  We show that the discrete Nahm equations are completely integrable in a
natural sense: to any solution we can associate a spectral curve and a
holomorphic line-bundle over the spectral curve, such that the discrete-time DN
evolution corresponds to walking in the Jacobian of the spectral curve in a
straight line through the line-bundle with steps of a fixed size. Some of the
implications for hyperbolic monopoles are also discussed.
