GLF4 AT FL410 EXPERIENCED TCAS RA WITH UNREPORTED TFC; COM WITH ATC FACILITY INDICATED SIMILAR CALL SIGN ISSUE MAY HAVE BEEN CAUSAL FACTOR.

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A TA APPEARED AT OUR 10 O'CLOCK POS. THE ACFT WAS IN A CLB 1200 FT BELOW AND 15-20 NM LATERALLY. WE LOOKED VISUALLY FOR THE ACFT. THE PNF MADE VISUAL CONTACT. APPROX 5 SECONDS AFTER THE TA; WE RECEIVED A 'CLB' RA. THE PNF STATED 'HE'S HEADED RIGHT AT YOU.' I DISCONNECTED THE AUTOPLT AND AUTOTHROTTLES; INSTRUCTED THE PNF TO INFORM ATC OF THE RA AND FOLLOWED THE TCAS CLB COMMAND. WHEN THE RA WAS ISSUED THE OTHER ACFT WAS APPROX 10 NM AND 600 FT BELOW OUR ACFT. UPON CLBING THROUGH FL417; WE RECEIVED A 'CLR OF CONFLICT' COMMAND FROM THE TCAS. I LEVELED THE ACFT AT FL420 UNTIL INSTRUCTED BY ATC TO RETURN TO FL410. INITIALLY; ATC WAS UNAWARE OF CONFLICT. UPON FURTHER INVESTIGATION ATC DISCOVERED THERE WERE 2 ACFT WITH SIMILAR CALL SIGNS ON THE FREQ; AND THAT 1 OF THE 2 ACFT WAS NO LONGER ON FREQ. A CONFERENCE CALL WAS MADE TO THE SUPVR OF THE ATC FACILITY THAT EVENING. WE WERE INFORMED THAT 1 OF THE 2 ACFT WITH SIMILAR CALL SIGNS ENDED UP ON THE NEXT CTR FREQ. THIS FREQ CHANGE WAS IN ERROR AND THE ACFT RECEIVED A CLB CLRNC FROM THE NEW CTLR. THIS FREQ CHANGE ERROR WAS THE CAUSE OF THE TA/RA. AT THE TIME THIS RPT IS BEING FILED; I DO NOT KNOW IF THE WRONG ACFT WAS GIVEN A FREQ CHANGE; OR IF WRONG ACFT TOOK A FREQ CHANGE FOR THE OTHER SIMILAR TAIL NUMBER. DILIGENCE AND DISCIPLINE MUST BE MAINTAINED ON THE PART OF PLTS AND ATC WHEN THERE ARE ACFT WITH SIMILAR CALL SIGNS ON THE SAME FREQ. THIS DILIGENCE AND DISCIPLINE INCLUDES THE ATC FACILITY RECEIVING THE ACFT VIA HDOF FROM THE BORDERING SECTOR. ACFT NEED TO BE INFORMED BY ATC OF THE SIMILAR CALL SIGNS BEING ON THE SAME FREQ; AND THE FULL CALL SIGNS SHOULD BE USED AT ALL TIMES BY THE CTLRS AND PLTS. THE CTLING ATC SECTOR THAT IS HANDING OFF THE 2 ACFT WITH SIMILAR CALL SIGNS; SHOULD INFORM THE NEXT SECTOR ABOUT SIMILAR CALL SIGNS WHEN XFERRING ACFT TO THE NEXT SECTOR.