EMB145 FLT CREW REPORTS OIL IMPENDING BYPASS ADVISORY MSG DURING CLIMB WHICH GOES AWAY. ON NEXT DEPARTURE MSG REAPPEARS AT 9500 FEET AND ENGINE FAILS.

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DURING DEP OUT OF ZZZ1 AN E1 OIL IMPENDING BYPASS MESSAGE WAS NOTED. WE REFED THE QRH ASSOCIATED WITH THIS MESSAGE. THE QRH STATED TO MONITOR ENG OIL TEMP AND PRESSURE (ALL WERE CONTINUOUSLY GREEN); AND IF THE MESSAGE DOES NOT EXTINGUISH TO CONTACT MAINT. SHORTLY AFTER THE DISPLAY OF THE ADVISORY MESSAGE; THE MESSAGE EXTINGUISHED FROM THE SCREEN AND WE CONTINUED TO ZZZ. DURING THE CLB PHASE OF THE FLT FROM ZZZ1 AT APPROX 9500 FT MSL WE NOTICED THE ADVISORY MESSAGE 'E1 IMP BYP' MESSAGE AGAIN. SHORTLY AFTER THE MESSAGE DISPLAYED ENG GAUGES QUICKLY STARTED TO FLUCTUATE AND ENG #1 FAILED. I DECLARED AN EMER WITH ATC AND WE REFED THE APPROPRIATE EMER PROCS AND QRH CHKLISTS FOR SINGLE ENG OPS. THE FLT LANDED SAFELY BACK IN ZZZ. NO PAX OR CREW MEMBERS WERE INJURED. THE FLT TAXIED TO THE GATE WHERE MAINT WAS INFORMED OF THE SITUATION. IN FUTURE SITUATIONS WHEN AN 'E1' OR 'E2 IMP BYP' ADVISORY MESSAGE IS DISPLAYED; MAINT WILL BE CONTACTED IMMEDIATELY TO REVIEW THE SITUATION EVEN THOUGH THE PROC DOES NOT REQUIRE IT. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: POST FLIGHT EXAMINATION BY MAINTENANCE REVEALED THAT A BURNER CAN WAS COMING APART AND THE PIECES DAMAGED THE TURBINE CAUSING THE ENGINE TO FAIL. HOW THE IMPENDING FILTER BYPASS MSG AND THE BURNER FAILURE ARE RELATED IS UNKNOWN. THEY MAY BE COMPLETELY SEPARATE INCIDENTS BUT THAT IS UNLIKELY GIVEN THE TIMING OF THE EVENTS.