I didn't know that this was a music calendar until I did some research. I found out a couple of things about this, though:
- The Lums and the group shots in the background are cels (Horus does not scan these by themselves)
- The backgrounds are interchangeable, and there's only four of them (Horus does not scan these by themselves)
- You can find the tracklist on Discogs (https://www.discogs.com/release/17074683-Unknown-Artist-%E3%81%86%E3%82%8B%E6%98%9F%E3%82%84%E3%81%A4%E3%82%89-Music-Calendar-1990)
- Once again, I can't find the music for this

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Taken from Horus Urusei Eye, a website that was dedicated to hosting Urusei Yatsura official art but has been offline for quite some time now. The site also hosted a bunch of other things such as art of other Rumiko Takahashi works, TV show theme songs, and the host's own weird fetish edits. These have not been preserved as well as the UY aspect of the site, so they may as well be lost to time.
If anyone has a larger backup of Horus Urusei Eye or has found it under another domain, please PM me!