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Super Smash Bros. Brawl
Palkia's first appearance in the Mario franchise is in Super Smash Bros. Brawl, in the Spear Pillar stage alongside Dialga and Cresselia. When the stage has a purple tint, Palkia has a chance of appearing. Once it appears, it has a multitude of abilities, including destroying sections of the stage, shooting a powerful beam either vertically or horizontally, decreasing the stage's gravity (making players jump higher and fall slower), tilting the camera so that it's askew, flipping it horizontally, and rotating it upside-down. After performing a couple of these abilities, it will disappear.
Super Smash Bros. for Nintendo 3DS / Wii U
In Super Smash Bros. for Nintendo 3DS / Wii U, Palkia can be spawned from a Poké Ball or Master Ball and attacks by using its signature move, Spacial Rend, which involves it slashing across the screen and flipping the camera. This affects nothing other than the player's viewpoint. Coming into contact with it does minor damage. Eventually, the screen tilts back to normal and Palkia disappears.
Blue indicates exclusive to the Wii U version.