Catalyzes the transformation of cystathionine into homocysteine. Also exhibits cysteine desulfhydrase activity in vitro, producing sulfide from cysteine. H2O + L,L-cystathionine = L-homocysteine + NH4(+) + pyruvate an S-substituted L-cysteine + H2O = a thiol + NH4(+) + pyruvate Amino-acid biosynthesis; L-methionine biosynthesis via de novo pathway; L-homocysteine from L-cystathionine: step 1/1. Homotetramer. Up-regulated at the transcriptional level when sulfate, cysteine, cystathionine or homocysteine are the sulfur sources. Repressed by methionine. Cells lacking this gene grow as well as the wild-type in the presence of sulfate, cysteine, homocysteine or methionine as sole sulfur source, but do not grow in the presence of cystathionine. Belongs to the trans-sulfuration enzymes family.