Brown Spots Above Ceiling/Ductwork
It looks like water damage to me.  Likely either a roof leak or a condensation problem with one of the ducts (or something else entirely that you don't know is there?).  It's also possible for the leak to be elsewhere above the ducts and this is just where the water found its way down to the ceiling.

Do you have access to the area above(attic)?  If so, maybe a couple pics  of what's there might help.

Also, you can buy a moisture tester help diagnose this.  If the ceiling at the stain is the same moisture content as surrounding ceiling, then the problem isn't currently happening but that doesn't mean it won't come back.  If it gets wetter during or immediately following a storm that almost guarantees a roof problem.