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Basically, after you defeat the Elite Four one of them will start flying around Hoenn. You will have to encounter it at random the first time, and from then on you can track it. You can get Latios in Ruby and Latias in Sapphire this way. Check that FAQ for even more detail.
To get the other in your game, you need the Eon ticket. That is given out at Nintendo events and on E-cards swiped through the E-reader.
i actually caught latios by accident. i originally fought him and injured him badly. he ran. when i found him again when i wasnt actually searchin he stayed at that health. i didnt know what to do, so i chucked an ultra ball. he wasnt even at miniscule health and i caught him :)
Okay this is the strategy to capturing latios and latias buy alot of ultraballs or use whatever you want and if you have a pokemon that can make the pokemon fall asleep use that but if you have a master ball yous that okay go to fortree city walk in the grass till you get to lilycove annd keep doing this look in all the grass and dont fly i'll repeat DONT FLY the'll disapear if you fly somewhere else walk all the way from mauville to get there when you walk throu the grass and go back throu routes it freezes latios and latias in there postion and gives you an easier way to catch them dont get mad after the first 15 minutes or so if it takes that long it took me 3 to 5 minutes to do this and if you wanna freeze him or her get repel and walk throu the routes then dont use anymore and then it will be more easier to find them this may take awhile but i know yall can do it and thats it.
eon ticket takes you to latias and you hafta find latios by luck
In sapphire,
eon ticket gets u latios and u hafta find latias by luck
the luck ones move a route to another route every time u enter a different city/route/group of routes.
DO NOT EXIT if you get to a southern island cause you'll never return there even with the eon ticket still at your disposal!!! [sorry for exaggerating but my friend screwed up, but good thing he saved before leaving].