Two statistically stationary states with power-law scaling of avalanches are found in a simple 1 D cellular automaton. Features of the fixed points, the spiral saddle and the saddle with index 1, are investigated. The migration of states of the automaton between these two self-organized criticality states is demonstrated during evolution of the system in computer simulations. The automaton, being a slowly driven system, can be applied as a toy model of earthquake supercycles.