Learjet Captain reported an engine overheat after takeoff during climb. The crew stabilized the overheat; received priority handling and returned to the departure airport.

Narrative

We departed out of ZZZ with passengers as a part 135 flight to ZZZ1. When climbing through FL230 a R PYLON OVHT message came on. Both Person 1 and I looked at the message and he asked if I would like him to grab the QRH; I said yes I will get the radios and fly the airplane. I institutionally reduced the power on the right engine when we got the message. He started through the checklist; the second line item was to reduce the power on the engine. I reduced the power to idle and the message continued to stay up. I contacted ATC and told them that we needed to return to ZZZ and that I was requesting priority handling due to an engine overheat and possible engine shut down. The message continued to stay on for the remainder of the checklist; Person 1 finished the checklist for R PYLON OVHT and the next step was to go to the engine shut down checklist. The message was illuminated and right when we were about to shut down the engine the message disappeared. I would estimate that it was on for at least 15 seconds after pulling the power on the right engine to idle. We had a brief discussion and decided to not shut down the engine but to still continue back to ZZZ. We let center know that we did not need them to roll the trucks and that the message was gone and we were flying with both engines for the moment. Person 1 ran the landing numbers and it showed that we were 500 lbs overweight for the landing; we both agreed it was in our best interest to continue and land immediately instead of flying around to burn fuel and risk bringing the message back and having to shut down the engine. We landed smoothly and pulled into the ramp without any further messages. From the moment I pulled the power to idle on the right engine; the N1 did not raise above 65% for the rest of the flight. The majority of the time it was at idle since we were descending into ZZZ.