My Breakout FXSB has fault code P0661 Intake Solenoid Low/Open. I replaced the intake valve and reset the code. Unfortunately, it keeps coming back. The engine runs fine, though. Does anyone have an idea?

 

I would suggest checking wiring circuit for any corrosion or high resistance. The code also suggest it could be in the wiring.
Good luck with search

 

When a code is displayed it doesn't mean the device is actually faulty.

You need to ensure the wiring is intact and the connectors are good. I would expect there to be a way to test the device described in the electrical diagnostic manual.

 

AIS is included on HDI vehicles only.
See Figure 6-33. The active intake solenoid (AIS) regulates
the amount of air entering the air cleaner.The AIS opens when
vehicle speed exceeds 45 mph (70 km/h) with 50% or greater
throttle opening. Once open, active intake will close when
vehicle speed falls below 40 mph (65 km/h). The power for the
AIS comes from the BCM. The ECM provides the path to
ground to trigger the AIS.