What determines a user library?

One of my customers has a library on their system that was created many many 
years ago that contains all their custom code and custom files. This library 
begins with the letter "Q" which is the first letter of the company that 
originally created the library. While I know itâs not a good idea to name a 
library beginning with the letter "Q", I donât know if thatâs part of my 
problem.

The problem is that this library does not show up as a user library on the 
system. When I run the command DSPLIB LIB(*ALLUSR), it does not show up. When I 
look in the Change Library Backup List off the SETUPBCKUP menu, it does not 
appear. I also can not do a SAVLIB *ALLUSR because it will not be included on 
the save. I am forced to save this library by name to get a backup on it.

What on the system is telling it that this is not a "User" library? And how can 
I change it?

V5R2 and I believe this library ad the same issue on previous releases.

Thanks for the help!

David Smith