FLC OF A DC9 LEVELED OFF DURING DSCNT IN RESPONSE TO A TCASII RA AND OBSERVED A TWIN CESSNA PASS UNDERNEATH THEIR ACFT ABOUT 600 FT BELOW ACCORDING TO TCASII.

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WE WERE JUST PAST FJC VOR ON THE MILTON 1 ARR (MIP.MIP1) TO LGA. WE HAD JUST RECEIVED A CLRNC FROM ZNY (135.0 FREQ) TO DSND FROM 12000 FT TO 10000 FT. AT APPROX 11400 FT WE GOT A TA ON THE TCASII. AT APPROX 11100 FT; WE GOT AN RA ON THE TCASII; THAT SAID TO STOP DSCNT. WE LEVELED AT 11000 FT AND SAW A CABIN CLASS TWIN PASS DIRECTLY BELOW US HDG NW. THE TCASII INDICATED HE WAS 600 FT BELOW OUR ALT. I CALLED ZNY TO TELL HIM OF THIS INCIDENT. HIS RESPONSE WAS THAT HE HAD VFR TFC AT 9500 FT BUT THAT HE WAS NOT TALKING TO HIM. WHEN I CALLED ZNY TO RPT THE INCIDENT; I IDENTED OURSELVES BY OUR 'ACR X' CALL SIGN. THE CTLR RESPONDED USING A CIVILIAN ACFT CALL SIGN; BUT WAS RELATING INFO PERTINENT TO OUR INCIDENT. I QUERIED ZNY AGAIN USING 'ACR X.' THE CTLR; AGAIN HE RESPONDED WITH A CIVILIAN CALL SIGN. THE THIRD TIME I CALLED I ASKED IF ZNY COULD READ 'ACR X.' THE CTLR RESPONSE WAS YES. I ASKED IF HE HAD HEARD MY XMISSION ABOUT OUR TCASII RA AND VISUAL SIGHTING. THE CTLR RESPONDED YES. I TOLD HIM HIS RPTED 9500 FT TFC WAS OVER 10000 FT. THE CTLR RESPONSE WAS TO GIVE US A FREQ CHANGE.