Feel free to exercise to a point where it doesn't disturb your sleep or worsen your symptoms. It depends on the state of your recovery. I didn't use to able to walk even 15 minutes without losing my sleep at night. Now I can almost exercise at will.
I wouldn't do it on an empty stomach since being hungry. It would just cause unnecessary stress. Also, don't ever do long duration workouts at high intensity.
Personally I combine very short and very intense workouts like 5 minute full speed sprints and long slow workouts like walking.