Back in November 2007 we reported that Asus was planning to expand the Asus EEE line to include Desktop PCs. One of these was actually on display at the CeBIT 2008.

The Asus EP20 has the size and also some similarity to the already announced Asus Nova P22.
The Asus EP20 would run the same Linux OS as the currently available Asus Eee PCs on an Intel Dual Core with 1.87Ghz. A 2.5 inch HDD would be used for storage instead of SSD.

The sign standing on the EP20 is a fun read. For instance it says: "Performance: Good Enough Performance" or "Small: 1/12 size & 1/5 weight of Tower PC."
I think it is safe to say the details of the Asus Eee PC desktop line is still up in the air for the most part.
More details on the French site matbe. Via Eeesite.