By 1200, the Swiss plateau comprised the dominions of the houses of Savoy, Zähringer, Habsburg, and Kyburg. Some regions (Uri, Schwyz, Unterwalden, later known as Waldstätten) were accorded the Imperial immediacy to grant the empire direct control over the mountain passes. With the extinction of its male line in 1263 the Kyburg dynasty fell in AD 1264; then the Habsburgs under King Rudolph I (Holy Roman Emperor in 1273) laid claim to the Kyburg lands and annexed them extending their territory to the eastern Swiss plateau.
Who claimed the Kyburg lands after their fall in 1264 AD?
the Habsburgs