Using the Aurora Ticket received from events on their copy of Pokémon FireRed & LeafGreen, or Pokémon Emerald games,
players gain access to the Birth Island, and get a chance to capture to Mythical Deoxys!
For players who rather not use the Mystery Gift to activate the event, simply use PKHeX's Event Flags > Research, and
For Pokémon FireRed & LeafGreen: Check Flag 2122 and 2123
For Pokémon Emerald: Check Flag 0314 and 2261
then proceed to add an Aurora Ticket into the Key Items of the Bag.
For Pokémon FireRed & LeafGreen, having the National Dex and Rainbow Pass may be required. More event flags here.
The shiny Deoxys here is obtained via Tool Assisted RNG.