Catalyzes the cleavage of L-cysteine to form 2-aminoprop-2-enoate and sulfide. The former then spontaneously hydrolyzes to pyruvate and NH(3). May be responsible for the production of sulfide required for the biosynthesis of iron-sulfur centers in this archaea. H2O + L-cysteine = H(+) + hydrogen sulfide + NH4(+) + pyruvate Binds 1 [4Fe-4S] cluster per subunit. Homotrimer. Belongs to the L-cysteine desulfidase family.