The　zero-error　capacity　of　a　noisy　channel　is　defined　as　the　least　upper　bound　of　rate　at　which　it　is　possible　to　transmit　information　with　zero　probability　of　error.　It　was　posed　by　Shannon　and　extended　to　channels　with　memories　by　Ahlswede,　Cai　and　Zhang.　In　this　paper,　we　give　a　first　step　towards　the　zero-error　capacity　problems　of　binary　channels　with　2-memories,　and　determine　the　zero-error　capacity　of　at　least　224　possible　cases　in　all　228　cases.　©　2024,　American　Institute　of　Mathematical　Sciences.　All　rights　reserved.