  We consider analytic, vacuum spacetimes that admit compact, non-degenerate
Cauchy horizons. Many years ago we proved that, if the null geodesic generators
of such a horizon were all \textit{closed} curves, then the enveloping
spacetime would necessarily admit a non-trivial, horizon-generating Killing
vector field. Using a slightly extended version of the Cauchy-Kowaleski theorem
one could establish the existence of infinite dimensional, analytic families of
such `generalized Taub-NUT' spacetimes and show that, generically, they
admitted \textit{only} the single (horizon-generating) Killing field alluded to
above. In this article we relax the closure assumption and analyze vacuum
spacetimes in which the generic horizon generating null geodesic densely fills
a 2-torus lying in the horizon. In particular we show that, aside from some
highly exceptional cases that we refer to as `ergodic', the non-closed
generators always have this (densely 2-torus-filling) geometrical property in
the analytic setting.
  By extending arguments we gave previously for the characterization of the
Killing symmetries of higher dimensional, stationary black holes we prove that
analytic, 4-dimensional, vacuum spacetimes with such (non-ergodic) compact
Cauchy horizons always admit (at least) two independent, commuting Killing
vector fields of which a special linear combination is horizon generating. We
also discuss the \textit{conjectures} that every such spacetime with an
\textit{ergodic} horizon is trivially constructable from the flat Kasner
solution by making certain `irrational' toroidal compactifications and that
degenerate compact Cauchy horizons do not exist in the analytic case.
