The Gyrannan system was a star system that was, in the mid-22nd century, located some distance from the influence of the Takret Militia.
In 2152, Rellus Tagrim and his comrades Guri and Renth planned to head for the Gyrannan system after leaving Enterprise NX-01, where they would be safe from the Militia. (ENT: "The Catwalk")
This system was only mentioned in dialogue.