You can only start the Eye of the Cosmos stages by completing Yukong Companion Mission. After completing the said mission, Xikui will message you for another errand. You will automatically obtain the Eye of the Cosmos after reading the message.
1) Finish battle within 10 Cycles(s)
2) Allies are knocked down for no more than 2 time(s)
3) Find the source of the unknown signal
➥ When you first arrive at Aureate Elixir Furnace, head to the huge furnace (right in front of you after spawning) to get a hidden achievement ‘I Was Here Flying: Aureate Elixir Furnace.’
➥ After that, continue flying further east from the furnace and you will see the starskiff on top of the roots.
➥ In the second zone, after you arrives, fly to the front a little near the huge tree, then look to your right to see two Mara-Struck Soldiers lounging on the rooftop. Take a picture of them to complete the suspicious signal objective.
➥ Look at the huge tree in the center again and you should notice a starskiff on a rooftop.
➥ When Xikui asks for help and you arrive at the 3rd area, head south towards an NPC vandalizing a cycrane. Take a picture of him.
➥ After that, just keep following him until he reaches the end of the jetty. You will enter a battle later after that, and there are 2 waves in it.