DC9 FLC AIRBORNE CONFLICT DURING SERIES OF HDG AND ALT CLRNC CHANGES NEAR BDF.

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WHILE LEVEL AT FL310; WE WERE TAKEN OFF OF OUR COURSE DIRECT TO IOW VORTAC AND GIVEN A 290 DEG HDG. WE WERE THEN GIVEN A DSCNT TO FL290 AND THEN A L TURN TO 270 DEGS AND AN EXPEDITED DSCNT TO FL290. THE FO ASKED WHERE THE TFC WAS AND THE RESPONSE WAS 2 O'CLOCK AND 12 MI AT FL310 SWBOUND. JUST AS WE WERE LEVELING AT FL290; TCASII PLOTTED A TARGET AT 11 O'CLOCK LEVEL AS A TA. IT TURNED ALMOST IMMEDIATELY INTO AN RA COMMANDING A DSCNT. I DISCONNECTED THE AUTOPLT AND BEGAN COMPLYING WITH THE RA WHILE THE FO CALLED TO INFORM ATC OF OUR ACTIONS. THEN THE PLT OF THE BIZJET CALLED US IN SIGHT AND ASKED ATC WHAT SHE WANTED HIM TO DO. THE BIZJET PASSED ALMOST DIRECTLY OVER US AT FL290 AS WE WERE PASSING FL287; 300 FT ABOVE US. WE CONTINUED THE DSCNT TILL TCASII ANNUNCIATED US 'CLR OF CONFLICT;' BY WHICH TIME WE WERE BELOW FL280. COMS FROM ATC WERE UNINTELLIGIBLE FOR ABOUT 3 CALLS WHILE WE ATTEMPTED TO VERIFY OUR CLRED ALT BECAUSE OF BACKGND VOICES. A MALE VOICE TOLD US TO MAINTAIN FL270. SUPPLEMENTAL INFO FROM ACN 458177: TFC INVOLVED WAS ACR A FLT XYZ WHO WAS AT THE TIME VECTORED ACCOUNT OF CPR JET X AT THE SAME ALT. I WAS COORDINATED FL290 WITH LOWER SECTOR WHICH WAS APPROVED. THIS WOULD ALLOW ACR A FLT XYZ TO GO BACK ON COURSE WHEN HE GOT LEVEL AT FL290. FL280 WAS NOT AN OPTION ACCOUNT ACFT B; NEBOUND J26 WAS DSNDING FL290 DSNDING TO FL240. I THEN PROCEEDED TO DO MY MANUAL DUTIES AND LATER I COULD TELL BY THE RADAR PERSON VOICE THAT SOMETHING WAS WRONG.