At the start of this chapter, you'll be at the bottom of an area after the battle with Castor. Walk up and press R1 to kill Pollux, then walk up to Castor and press R1 to grab the Amulet of Uroborus from his body. With this in your possession, hold L2 and press Square to target the broken bridge, using the right analog stick to start rebuilding it. Only rebuild it halfway, so you can use it as a stepping stone to the walkway on the left.
Jump up to the walkway on the left and grab the Red Orb Chest on the southern end, then go back and rebuild the bridge until it's whole again. Now you can jump up, but before meeting with the Oracle, go to the left and fix the Decayed Chest at the end of the path (Chest #1). Go to the Oracle to initiate a cutscene.
After the scene is over, head forward and jump the gap to a ledge. On the first ledge, go all the way to the right to find a Phoenix Feather. Go back and jump to the second ledge, then to the walkway on your left. Keep going forward until you can jump off, then go forward through the open gate to fight some enemies.
You must use the Amulet of Uroborus at least once in the battle, as the enemies will keep respawning if you don't. Once they're dead, go down the next path and heal the bridge at the end, following the new path to more enemies. Take these out and then heal the next part of the aqueduct. Go forward to arrive back in The Cistern.