According to the myths, cockatrices were supposed to fly ("The cockatrice was also said to fly using the set of wings affixed to its back", Wikipedia).

How is it in this game, cockatrices don't count as flying animals?

I have a quest here, requirement is flying animal with agility of 45. My cockatrice Zelda has exactly 45 agility but the quest page says that I don't have a pet that meets the criteria.

A glitch?

Being a hybrid between a rooster and a snake, they have only rudimentary wings and don't fly. If to get some parallels with reality, Cockatrices are very close to the late theropoda dinosaurs. They have feathers but don't have well-developed wings.

So cockatrices do not fly in Beast Keepers, that's an information.

(In the myths it seems they however did if Wikipedia is not wrong here...)

I apologize for the confusion. Our game has sci-fi back, not only fantasy :) Yes, in this case we stick to small raptor dinosaur type of creature which doesn't fly.

We don't have a native english speaker in our team right now and it's quite hard for us to write down all of the concept aspects. Probably such items like species back story will be written and published only after game release.