Date: 202101
Local time of day: 1801-2400
Location: ZZZ.ARTCC, US
Aircraft: Commercial Fixed Wing
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Climb
Airspace: Class E ZZZ
ATC / advisory: Center ZZZ
Aircraft component: Engine
Component problem: Malfunctioning
Reporter organization: Government
Reporter function: Enroute
Reporter qualification: Air Traffic Control Fully Certified
Human factors: Confusion; Workload; Time Pressure; Communication Breakdown; Situational Awareness
Anomaly: Aircraft Equipment Problem Critical; Airspace Violation All Types; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy
Detector: Person Air Traffic Control; Person Flight Crew
When detected: In-flight
Result: Air Traffic Control Issued New Clearance; Air Traffic Control Provided Assistance; Flight Crew Landed in Emergency Condition; Flight Crew Requested ATC Assistance / Clarification; Flight Crew Returned To Departure Airport
Contributing factors: Aircraft; Airspace Structure; Manuals; Procedure
Primary problem: Aircraft

Narrative: Aircraft X checked on and I climbed the aircraft to 23;000 feet; and gave them normal speed leaving 13;000 feet. A little bit later; Aircraft X and an aircraft on another frequency of mine transmitted at the same time. I responded to the other aircraft first; and when I went to look at Aircraft Xs data block; they were squawking XXXX. They requested to level at 12;000 feet and fly a 360 heading. I immediately assigned it to them. I called Departure; requested control and told them they were a priority handling. I went back to Aircraft X and they requested a 090 heading. I assigned that to them; notified my Controller in Charge; and began to collect the information. I asked for the nature of their situation; which they told me there was high vibration in their left engine.I also gathered that they wanted to return. It was difficult to understand the pilot due to a language barrier; but finally I understood how many souls on board there were and that there was more than 10 hours of fuel remaining. I then called Departure to forward the information and ask them for a heading and altitude I could turn the aircraft to. They gave me a 160 heading and 12;000 feet; as well as a frequency. After going back to the pilot; I called Departure back to ensure he was radar contact with them. Departure said yes; and asked if he needed to hold. I told them I had not asked. I then shipped the aircraft to TRACON. By the time the aircraft had turned back and I had shipped the aircraft; the aircraft did fly through and leave the southern tip of an adjacent Approach's airspace. However; I did not have a chance to call them as I was busy coordinating with the other TRACON and trying to get information from the pilot. I also never cleared the aircraft back to its departure airport. I simply turned the aircraft to a 160 heading and shipped the aircraft to Departure.I don't believe there was much I could do to prevent this situation. If I had the chance; I could have called TRACON; but in this scenario I didn't have any time to call due to the high workload.




Synopsis: A Center Controller reported they vectored an aircraft under priority handling and requested clearance to return to the departure airport through a TRACON's airspace without a point out.