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We describe the extremal translation invariant stationary (ETIS) states ofthe facilitated exclusion process on $\mathbb{Z}$. In this model all particleson sites with one occupied and one empty neighbor jump at each integer time tothe empty neighbor site, and if two particles attempt to jump into the sameempty site we choose one randomly to succeed. The ETIS states are qualitativelydifferent for densities $\rho<1/2$, $\rho=1/2$, and $1/2<\rho<1$, but in eachdensity region we find states which may be grouped into families, each of whichis in natural correspondence with the set of all ergodic measures on$\{0,1\}^{\mathbb{Z}}$. For $\rho<1/2$ there is one such family, containing allthe ergodic states in which the probability of two adjacent occupied sites iszero. For $\rho=1/2$ there are two families, in which configurations translateto the left and right, respectively, with constant speed 2. For the highdensity case there is a continuum of families. We show that all ETIS states atdensities $\rho\le1/2$ belong to these families, and conjecture that also athigh density there are no other ETIS states. We also study the possible ETISstates which might occur if the conjecture fails.

Stationary States of the One-Dimensional Discrete-Time Facilitated Symmetric Exclusion Process