ZSE CTLR EXPERIENCED OPERROR AT APPROX FL340 FORGETTING THE FACT THAT AN ACFT INVOLVED WAS NOT RVSM EQUIPPED.

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I WAS WORKING ACFT #1 NBOUND TO BLI AT FL340. ACFT #2 A HAWKER HUNTER WAS HANDED OFF TO ME FROM LOW ALT SECTOR 5 UP TO FL230 H10.J15.IMB..TWF.BOI. I WAS WORKING HIGH ALT SECTOR 16; ONE MAN; NORMAL TFC. I POINTED OUT ACFT #2 TO AN ADJOINING HIGH ALT SECTOR FOR CLB AND THEN CLBED ACFT #2 TO FL350. I DID NOT RECOGNIZE THAT ACFT #2 WAS NON-RVSM. HE WAS A CIVILIAN PLT FLYING TO HIF. HE WAS IMPROPERLY USING A MIL CALL SIGN. HE SHOULD NOT HAVE FILED INTO RVSM AIRSPACE; AND WHEN CLRED INTO RVSM AIRSPACE; HE DID NOT SAY 'NON-RVSM; CLBING TO FL350.' INSTEAD HE SAID 'LEAVING 230 FOR 350.' I RECOGNIZED THAT ACFT #2 AND ACFT #1 COULD BE A PROB; SO I CLRED ACFT #2 DIRECT TWF TO GO BEHIND ACFT #1. I THEN SEQUENCED 3 PDX ARRS. AFTER THAT SEQUENCING (8-9 XMISSIONS) I RETURNED TO THE ACFT #1 AND ACFT #2 SITUATION AND ASKED ACFT #2 HIS RATE OF CLB. HE SAID 1000 FPM. I ASKED HIM TO INCREASE TO 1500 FPM AND CALLED TFC ACFT #1. HE LEVELED AT FL350 ABOUT 15 NM W OF ACFT #1. I CALLED TFC TO ACFT #1 SAYING '17 MI W AT FL350.' ACFT #2 FLEW BEHIND ACFT #1 AND 2 HITS INSIDE 5 NM BUBBLE 4.77; 4.73 NM AND 1100 FT ACCORDING TO ZSE DATA. I DID NOT RECOGNIZE THAT ACFT #2 WAS NON-RVSM. MY DATA BLOCKS WERE SET TO #2; I WILL INCREASE THIS SIZE TO 4 AND TURN UP MY FDB BRIGHTNESS SO THE CORAL BOX AROUND THE 'C' IS BRIGHTER. I ONLY REALIZED NON-RVSM WHEN CONFLICT ALERT FLASHED ONCE WHEN THEY WERE PAST EACH OTHER; AT THE POINT OF ERROR. NO URET RED ALERTS ON THESE TWO; NO CONFLICT ALERT. THESE WERE ALL CONTRIBUTING FACTORS. I WAS HEALTHY AND ALERT -- NO CONTRIBUTING HUMAN FACTORS ON MY PART LIKE SICKNESS; TIRED; ETC AND NO DISTRS. I WRONGLY ASSUMED HE WAS MIL FLT AND RVSM.