THE FLC EXPERIENCED A 767-300 ENG FAILURE 340 MI SW OF BERMUDA AND DIVERTED FOR EMER SINGLE ENG LNDG AT NEAREST SUITABLE ARPT WHICH WAS BERMUDA.

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I WAS ON BREAK AT ABOUT XA20Z AND WAS CALLED TO COCKPIT. CREW HAD EXPERIENCED (R ENG OIL FILTER BYPASS) EICAS MESSAGE. THEY HAD COMPLETED I-18 (OIL FILTER) PROC. I ENTERED SEAT; REVIEWED PROC; AND FINISHED BRINGING THROTTLE TO IDLE. WE REQUESTED LOWER DUE R ENG AT IDLE (FL290). COM RADIO (ATC VIA ARINC) RESPONDING SLOW SO I DECLARED AN EMER AND STARTED DSCNT TO FL250-FL210. WITH COMPANY DISPATCH AND MAINT WE AGREED (2 ENGS RUNNING) TO DIVERT TO SJU. REQUESTED DIRECT SJU FROM ARINC. BEFORE WE RECEIVED CLRNC; R ENG QUIT SO I SAID TO TELL THEM WE'RE PROCEEDING DIRECT TO BDA (NEAREST SUITABLE). ATC CAME BACK WITH DIRECT BDA AND BLOCK ALT FL550 TO FL210 OR FL250. APU STARTED IMMEDIATELY AFTER ENG FAILURE. DUMPED CTR TANK APPROX 13000-14000 LBS. DSNDED TO FL150 TO BURN MORE FUEL TO TRY AND GET WT DOWN BELOW 320000 LBS (MAX LNDG GROSS WT) AND BETTER PERFORMANCE IF GAR ON 1 ENG. ENRTE WE HAD L PACK OVERHEAT WITH LOSS OF PRESSURIZATION UNTIL COPLTS REESTABLISHED PACK OP. MY REACTION WAS TO DSND TO 8000 FT WHILE THEY WORKED ON PROB TO KEEP CABIN BELOW 10000 FT (NO CREW OXYGEN MASK; ETC). WE VECTORED AROUND SOME WX ENRTE TO BDA. MADE NORMAL SINGLE ENG APCH AND LNDG. LNDG GROSS APPROX 317000-318000 LBS. BDA TWR SAID NO INBOUND TFC SO WE CHOCKED NOSEWHEEL ON RWY AND LET BRAKES COOL (APPROX 30 MINS); TAXIED TO GATE. ALL CHKLISTS COMPLETED INCLUDING R ENG OIL FILTER; ENG FAILURE; SHUTDOWN FUEL DUMP; SINGLE ENG APCH AND LNDG. CHK LIMITS FOR BRAKES ON RWY. WE KEPT CABIN AND PAX INFORMED. COMS: APPARENTLY WE WERE NOT EXPLICIT ENOUGH WITH ZNY AS TO OUR EXACT SIT. I PERSONALLY WAS NOT ON THE RADIO. THE ACFT WAS NEVER IN ANY DANGER. A RADIO XMISSION APPARENTLY WENT OUT WHEN WE LOST THE ENG TO 'SEND THE HELI.' (THE EXTRA COPLT MAKING THE CALL THEN ASKED ME -- I BALKED AT 'HAVING THEM IN THE AIR.' HE THEN ASKED IF I WANTED THEM 'READY TO GO.' I SAID 'THAT SOUNDS LIKE A GOOD IDEA'.) I HAVE ONLY A VAGUE RECOLLECTION OF ABOVE CONVERSATION BUT THE COPLT IN FRONT SEAT REMEMBERS IT THIS WAY. THE EXTRA COPLT THEN IMMEDIATELY RESCINDED HIS CALL FOR HELIS (WITHIN 30 SECONDS; ON TRANSCRIPT). ZNY ASKED SEVERAL TIMES IF WE WERE GOING TO MAKE BDA (I'M NOT ON FREQ). EXTRA COPLT SAID 'YES.' HE KEPT THEM ADVISED OF DSCNT TO FL150 BUT (MUST NOT HAVE MENTIONED TO 'BURN MORE FUEL') WE WERE IN CONSTANT COM WITH COMPANY THROUGH SATCOM. ONE GOOD EXPLANATION TO ZNY WOULD HAVE CLRED UP SIT OR WHY DIDN'T THEY ASK? THEY DIDN'T ASK WHY WE WERE AT FL150. THERE IS A POSSIBLE JOB ACTION INVOLVED HERE BY ZNY. I DON'T KNOW FACTS YET. THE EXTRA COPLT WAS AN INVALUABLE ASSET ON THIS EVENT. HE HANDLED MUCH OF RADIOS AND COM WITH CHIEF PURSER AND ALL CHKLISTS. KEEP THE THIRD MAN!