Byeol

When it appears

Once you've completed the main quests Grandmaster Promotion and Destroy Vados and then talked to the Admin in Guild Lobby, in Guild Entrance, it can turn up in the wild. It stops turning up once you own a Byeol, unless you've completed the main quest Destroy Vados and taken Eliza up on her offer in Hilda's Room.

Capturing

It has no favourite foods, so each food adds 1 to your capture chance, and Perfect Berry 10. As a Legendary Nexomon, it's harder to capture: food counts for less, Nexotraps count for half, and wearing it down adds at most 10 instead of 20. A Golden Nexotrap still adds the full 100. Catching a Nexomon explains how the whole chance adds up.