You can't catch shiny legendary Pokemon without cheats... sorry
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you cant
If your legendary Pokemon run away in Shiny Gold, there is no way of retrieving them again.
In order to catch a shiny rayquaza in any game without cheating is to save right in front of any legendary and keep battleing it. I recommend using a masterball unless you are EXTREMMLY GOOD at catching Pokemon, It will take you about 14 hours to see one. Your chances of finding one are about 1 out of 8164. I migrated all of my good Pokemon in sapphire to platinum just to start a new game to catch a shiny kyoger, rayquaza, regice,regirock,registeel, and If you are BEOND lucky Latias(ITS YELLOW NOT RED!!!) Have good shiny legendary hunting:P! JB4374! Also if your lucky you might get a shiny latos, it's green!
you have to get more then 45 gym to catch legendary Pokemon! deoxys is a legendary Pokemon
It takes luck to come across a shiny, but when you find one you catch it like you would catch a regular Pokemon.