Well, you know about when they go back to the meeting room in disk two? And there are two doors to take? One leads to an underground reservoir and the other leads to the Fire-Wall dungeon. I took the reservoir thinking that's where I had to go. The path leads to another section of the ruins that looks just like the first and also incorporates the use of chessmen. There is a boss to get to the second section at the end of the reservoir. There is a boss for the second section, It's a that kid wind spirit you met at that one creepy dungeon (I'm sorry I forgot the name, but it's where they find those weird test samples), and two other elemental bosses. THEY WERE HARD. After that, there is a weird spirit in the next room who plays you at chess. After you beat him he mentions that you might be able to find "his place". The rewards are a few treasures.. I don't know if this is a secret dungeon or not. Sorry that I can't be too specific. It's been a while since I've played the game.
Have you done any secret dungeons?
EDIT: Also, feel free to join the Star Ocean fan club group I've created ^__^! No one has joined (except for Zajice, but he does not post anything). If you want, you can invite other people you know who like Star Ocean ^__^! You're the only one of my friends who knows about the series ^__^!