Thurston Plunkett said "check the Solarium and the Barbican, if you can gain entry."
Since this is a Side Quest, it is possible that you do not have access to those areas yet. This happens a lot in the game. We get a Side Quest (like the ones from our teachers) to go to an area that we have not opened yet.
If that is the case, just continue to follow the Main Quests in the storyline, and they will eventually lead you to the Solarium and the Barbican.
In case you need more information, this link shows all of the quests in Celestia (Main, Side, Crafting, etc.):