Yes the same way you get shiny Pokemon in the other Pokemon games is how you get shiny Pokemon in heartgold. There is a free shiny Pokemon in the lake of rage.
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Luck. You have a 1 in 8,192 chance of your starter Pokemon to be Shiny.
I believe you can get a Shiny Stone in HeartGold in the Pokèathlon Prize Corner after you get 5000 Points.
To get any Shiny Pokemon, you need: LUCK
i am fairly sure you can't.
You can get them by talking to your Pokemon while walking.