  Which relativistic field theories give rise to Kepler dynamics in the
two-body problem? We consider a class of Hamiltonians that is the unique
relativistic extension of the Kepler problem preserving its so(4) algebra, and
have orbits related through time reparametrisation to orbits of the original
Kepler problem. For three explicit examples, we give a natural interpretation
in terms of spin-0,-1 and -2 interacting field theories in 5D. These are
organically connected via the classical double copy, which therefore preserves
maximal superintegrability.
