Flight Instructor reported the need to take evasive action on downwind to avoid a traffic conflict.

Narrative

Myself; and my Student were in the left downwind for Runway 36 at ZZZ. I was sitting in the left seat per school lesson requirements. Abeam our touchdown point we began a descent and began configuring for landing. As we reached a lower altitude which I cannot fully recall; approximately 1600-1700 MSL; I looked forward and saw what appeared to be an aircraft descending directly in front of us; on an eastern heading. As if they were approaching a base leg. I took the controls immediately and took action to avoid the aircraft; banking us to the right. This was our second lap in the pattern at ZZZ; and we had been making continuous position reports throughout; and had not heard anything from any other traffic in the area. I made a call to the traffic to let him know we were avoiding; to which there was no response. Still under control of the aircraft; I climbed us up to 2500 MSL; and instructed my Student to observe the traffic; He watched them land and take off; departing the area to the east. He then looked on software and found their tail number. Upon further investigation; neither myself; nor the Airport Manager; where we believed this aircraft to be based at; could find the Owner or Pilot flying the aircraft. I believe had I not seen the aircraft and taken immediate action; we were going to collide sometime in the base to final portion of our flight.