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. Is very specific for L-cysteine, with no activity being detected with D-cysteine, L-homocysteine, 3-mercaptopropionate (cysteine without the amino group), cysteamine (cysteine without the carboxylate), or mercaptolactate (the hydroxyl analog of cysteine). H2O + L-cysteine = H(+) + hydrogen sulfide + NH4(+) + pyruvate Binds 1 [4Fe-4S] cluster per subunit. kcat is 39.6 sec(-1) (at 65 degrees Celsius). Optimum pH is 7.6. Homotrimer. Belongs to the L-cysteine desulfidase family.