When you enter the canyon, the plants at the entrance mention that a lion cub entered the canyon a little bit ago. Head south and then climb down the vine. You'll eventually fall off the end of the vine and fall down to the bottom of the canyon. Head west.
There's Liem, but he's cornered by a couple of birds. Destroy the birds and talk to Liem. Liem is glad to see someone and thanks you for rescuing him. Ark realizes that this canyon is too dangerous for just a small lion cub to be in all by himself, so he decided to accompany Liem through his ritual.
When you enter the new area, go all the way to the east side of the cliff. Liem will be waiting for you to talk to him. Liem will then head back to the west. Follow him and you'll end up jumping up more ledges to the entrance into the next area. Remember not to exit the area before Liem does, or he will stay behind and you'll have to go back and talk to him again before he starts moving back up the cliff.
Liem will walk west until he gets to a rock-path maze. He'll end up going the wrong way into a dead end, so talk to him and he'll start walking in the right direction. Start through the rock maze and he'll stop again half way. Liem will be unsure as to which direction to take, left or right. Tell him to go to the right. The two of you will continue on through the maze and head west again.
In the next area Liem wants to race you up the cliff. Even if you start climbing up the cliff before he says "go", he'll still beat you to the top. The two of you head west. In the next area, Liem says he senses a monster up ahead and to be careful. He jumps the ledge across to the other side, but you can only climb down the vine and walk west into the next area.
The next area has you and Liem separated. Liem will be on the ledge above you, and there will be some sort of mud monster on the other side of the gap. The Mud Doll says he wants to "party" with you, and will start spitting out globs of mud at you. There will be a rock right below Liem. Pick it up and jump/throw the rock at the Mud Doll. Liem will go off the screen and push another rock back down onto the ground, so while you're waiting for him try to avoid the globs of mud. The Mud Doll only has about 100HP, so once he's damaged he'll apologize for being mean and give you a gift.
He'll create a bridge for you to walk across the gap onto his side. About half way over the bridge though he'll laugh at you, destroy the bridge, and you'll go falling down into the gap, landing on a small ledge with no way back up to the top. From up top, you can hear Liem yell for you and the Mud Doll laugh. Liem will get mad and all you can hear is something fighting. After a while everything's quiet up top, and then Liem will come jumping down the cliff to see if you're still alive. He's happy to see you, and he is proud to announce that he destroyed the Mud Doll. A shiny stone falls from up top, and Ark tells Liem to keep it as a token of his challenge. Liem receives the Starstone. He also says that he's going to get a Gull to help fly Ark out of the Canyon. The gull shows up and fly's back to the Safarium.