Once you get the silver wing from the director of the radio tower who was kidnapped by team rocket then when he's saved by you he will give it to you then teach a Pokemon the HM whirlpool then fly to cinnawood city then go east you will find a cave actually there's multiple caves one has the entrance to a cave where you can use the HM waterfall go in there and use the HM you will soon find a cave in the middle of nowhere in there is lugia. Weaken it to red hp then paralyze it then throw many ultra balls eventually you will get it.
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go to whirl islands
You can't get Dark Lugia in Pokemon Silver, Gold, or Crystal. You can get Dark Lugia in Pokemon XD though. For Pokemon Silver, Gold, and Crystal it is just the regular Lugia.
In the whirl islands.
The best way to catch the Lugia is through the Caves. You must have a silver feather before entering. Navigate the path until reaching the waterfall where the Lugia is located then engage in a fight using your best level 40 or higher Pokemon.
you go to the third gym you go up you go to the grass and you randomly walk around and voila! dark lugia if you want shiny do the same thing but release your mew and mewtwo and there is shiny lugia.