  We study stationary stable processes related to periodic and cyclic flows in
the sense of Rosinski [Ann. Probab. 23 (1995) 1163-1187]. These processes are
not ergodic. We provide their canonical representations, consider examples and
show how to identify them among general stationary stable processes.
  We conclude with the unique decomposition in distribution of stationary
stable processes into the sum of four major independent components: 1.
  A mixed moving average component. 2. A harmonizable (or ``trivial'')
component.
  3. A cyclic component 4. A component which is different from these.
