AN ACR B767 FLC PERFORMED AN INFLT SHUTDOWN OF THEIR L ENG AND DIVERTED TO MAO FOR AN OVERWT LNDG. THE EMER; DIVERSION AND LNDG WENT WELL; BUT IMMIGRATION INSISTED THAT THE PAX HAD TO REMAIN ON BOARD FOR OVER 2 HRS AFTER ARR AT THE GATE WHILE THEY PROCESSED OTHER FLTS. MEANWHILE; THE CFR PERSONNEL FELT THAT THE BRAKES HAD OVERHEATED AND THEY FILED BRAKE FIRE RPT. THE FLC DISPUTES THE RPT.

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DEPARTED GRU FOR MIA. AFTER OVER 3 HRS AT CRUISE; GOT 'L ENG OVERHEAT' AND 'EICAS DISAGREE' LIGHTS. FOLLOWED OPS MANUAL PROC. ATTEMPTED TO CONTACT MAINT SVC VIA FLT SUPPORT PHONE PATCH TO DISCUSS MEANING OF COMBINATION OF EICAS MESSAGES. UNABLE TO COMPLETE PHONE PATCH DUE TO POOR RADIO QUALITY. RELAYED INFO TO DISPATCH AND OTHER ACFT. REQUESTED DIVERSION TO MAO FROM CTR; SHUT DOWN L ENG; DECLARED EMER. OVERWT; SINGLE-ENG LNDG AT MAO WITH CFR STANDING BY. TOUCHDOWN; ROLLOUT AND TAXI TO GATE WERE NORMAL. CFR FOLLOWED US TO THE GATE AND STOOD BY TO MONITOR ACFT. MAO IMMIGRATION DECLINED DEPLANING PAX UNTIL OTHER FLTS HAD CLRED CUSTOMS. IT TOOK OVER 2 HRS TO DEPLANE PAX AND CREW. WHEN I RETURNED TO THE GATE FROM OPS; I WAS INFORMED THAT THE CFR CREW TOLD THE FO AFTER THE FACT THAT THE BRAKES HAD OVERHEATED. THE FO IMMEDIATELY WENT TO CHK AND FOUND THEM WARM; BUT NORMAL. THERE WAS NO NEED TO CONSIDER AN EVAC. I WAS INFORMED TODAY THAT THE MAO ARPT AUTHS WERE FILING A 'BRAKE FIRE' RPT. OUR MECHS FOUND NO EVIDENCE OF A FIRE. AFTER LNDG; TO CONSERVE BRAKE ENERGY; USED THE ENTIRE RWY; SLOW TAXI; ETC. I AM CERTAIN THAT THERE WAS NO BRAKE FIRE. SUPPLEMENTAL INFO FROM ACN 383477: AFTER ALL DUTIES COMPLETE PROCEEDED OUTSIDE TO INSPECT ACFT. TALKED TO CFR THROUGH INTERPRETER; VERY DIFFICULT TO INTERPRET WHAT HE WAS SAYING. WAS ABLE TO DETERMINE THAT ALL WAS SAFE AND WELL AT THIS POINT IN TIME. INTERPRETED FROM HIM THAT WHEN WE ARRIVED AT GATE THAT OUR L LNDG GEAR WAS VERY HOT. I THEN INSPECTED BOTH LNDG GEAR AND WAS SATISFIED SIT WAS SAFE AND SECURE. I REQUESTED THAT CFR REMAIN ON STANDBY UNTIL FURTHER NOTICE; RETURNED TO ACFT AND UPDATED CAPT ON SIT AND ASSISTED WITH HANDLING OF PAX.