The Axew encounter is located in the middle of Route 3. Only accessable once the player has received the ability to Rock Climb.

After going down there will be a cave inside of which will be 3 Mining Rocks a Dragon Fang and a potion hidden inside a rock in the North-East Corner.
After interacting with the egg and choosing to take it, it's mom and dad will show up to have a double battle with you, they will both be Lvl 93.

After defeating them they will go away leaving you with the egg.

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