1. Obtain the Eon Ticket:
a. Beat the Elite Four and become the Pokémon Champion.
b. Go to the Space Center in Mossdeep City.
c. Talk to the scientist on the left to receive the Eon Ticket.
2. Travel to Southern Island:
a. Go to the dock in Lilycove City.
b. Talk to the sailor and select "Southern Island."
3. Find Rayquaza:
a. Enter Sky Pillar.
b. Climb to the top of the tower and enter the chamber at the summit.
c. Rayquaza will appear.
4. Save your game before battling Rayquaza.
5. Weaken Rayquaza:
a. Use Pokémon with super-effective moves against Dragon-types, such as Ice-type moves.
b. Bring along Pokémon that can lower Rayquaza's HP without causing it to faint.
c. Be prepared for long battles. Rayquaza has a high HP and stats.
6. Throw a Dive Ball or Ultra Ball:
a. Dive Balls have an increased catch rate for Pokémon encountered while underwater.
b. Ultra Balls have a high catch rate overall.
c. Keep throwing Dive Balls/Ultra Balls until Rayquaza is caught.
7. Continue resetting until you encounter a shiny Rayquaza.
Note: Shiny Pokémon have a very low encounter rate, so it may take many attempts before encountering a shiny Rayquaza.