This page explains how to do the Sunken Desert sidequest.
How to Start the Quest
For this quest, go to 600 A.D. and go to Fiona's Villa (north of Porre). You should see a swirling hole near her house. If not, go to 12,000 B.C., go to the Commons, and talk to the woman with the sapling and tell her to plant it. Then go back to Fiona's Villa, and the swirling hole should be there.
Go to the Sunken Desert
The recommended party is Frog, Robo, and Ayla. Equip them with Shadow-absorbing armor, as well as status-protecting gear, such as the Vigil Hat, Sight Cap, and Amulet.
Check on the swirling hole near Fiona's Villa to go in, and walk north into the center of the vortex to be drawn in.
You can ignore the chests in the area for now, because they are guarded by monsters that will go away when you defeat the area's boss (unless you want the experience points, skill points, etc. from the battles).
Since the desert here is slippery, you will need to run in order to move around. Go north of the nearby chest, and west to the solid ground just below the stairs (the stairs will take you back outside if needed). Go south and you will have to fight. Have Frog use Water2, then use physical attacks. Continue going south, and go down the stairs.
In this next area, each time the floor shakes, you lose 30 HP, so quickly go around and find the boss that keeps appearing in random locations.
Retinite
You can Charm a Speed Tab from the Retinite Core. First use Water 2 to weaken all three parts of Retinite, then use physical attacks and single-target physical Dual Techs such as Beast Toss on the top half. If the core dies, then the bottom and top's defense will go up with each hit. If you brought a party member with non-Water magic, you can heal the core with non-Water magic attacks to prevent the core from dying. Once you kill the top, attack the bottom with physical attacks and single-target physical Dual Techs such as Spin Kick (since Beast Toss doesn't work on it). If you kill the bottom before the core, you can try to quickly kill the core with physical attacks to get more experience points, but it will soon run away, and you might not be able to kill it in time.
Sunken Desert
Now go around the room and open the seven chests to get a Full Tonic, 5000 G, a HyperEther, an Aeon Helm, a MuscleRing, a Memory Cap, and a Full Ether. Then go to the northwest corner of the room and go through the cave opening to return to the first room.
There is a Power Tab to the left of the center of the room. Open the four chests to get an Elixir, a Full Ether, a Lapis, and an Aeon Suit.
Use the north exit to go back outside. Put Robo in your party, then go to Fiona's Villa and talk to her. Agree to let Robo stay behind.
Go to 1000 A.D. Save the game in case you make a mistake in the next part, so you can try again.
In 1000 A.D., go into the center structure, Fiona's Shrine, and talk to the nuns if you want to buy some useful Helms. If you haven't saved the game yet, do so now. Then go to the north part of Fiona's Shrine and check on Robo.
Watch the cutscene. When Lucca awakens, go southeast. Go through the Gate.
Don't go through the southwest door. Instead, go down the stairs. You can optionally go north through the door and read the note for a clue on what to do next. Then leave that room and go south through the door below. Then when the machine turns on, go as far south as you can go, then as far east as you can go, then press A, then when it says "Enter the password," press a button to dismiss the dialog. Then press L, A, R, A (Lucca's mom's name, which is on a piece of paper in a room above). If you don't enter the code in time, you will need to load your saved game if you want to try again.
Afterward, you can read the papers on the floor if you wish. Go through the Gate.
Talk to Robo to get the GreenDream, which is an accessory that will automatically revive the wearer if they are killed, once per battle.
If you stopped the machine, you can visit Lucca's house and talk to everyone there to see what has changed.