You can actually place the ring on any of the desks, and you have to figure out which desk is the correct one. You can figure it out on your own based on clues within the game, but if you don't want to it's the room with the rugby ball on the bed. After that you have to finish the game again by opening the pod, and there will be an extra line of dialogue in the ending. After that the Alyssa story will be unlocked on the main menu.
The extra content is actually something that differs between the English and Japanese versions.
This tells a story about how the leeches got aboard Damrey. Large brought a fossil from the alien planet on-board the spaceship with the intent of selling it when he got home, but when he accidentally spills water on it it comes to life and attaches to his arm. He tries to get it off, but only manages to split it into more leeches. He eventually (accidentally) manages to re-fossilize them with a disinfectant, but later another crew member accidentally un-fossilizes them again. It's pretty inconsequential, but it offer more of a look into the personalities of most of the crew members.
From what I've read this scenario is about Alex, the only crew member you don't ever encounter in the game. Told from her perspective while in Cold Sleep, Alex wakes up from a dream in which she gave birth to an alien. She then complains about stomach cramps, and then something appears to come out of her stomach.
I'm pretty upset about this difference because the Japanese story has huge ramifications on the game's plot, and without it we never find out what Alex's fate is. It was probably cut for rating reasons.