ACR X TCASII RA HAD LTSS FROM ACR Y. SYS ERROR. EVASIVE ACTION TAKEN.

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ACR X TOOK OF AND WAS EXECUTING THE LGA 4 DEP RWY 13 (MASPETH CLB). WE WERE ON DEP FREQ 120.4. WE HAD REACHED 3000 FT AND MADE OUR R TURN TOWARDS 340 DEGS. WE HAD BEEN RECLRED TO 6000 FT. WE WERE GIVEN HDG CHANGES TO 320 DEGS FOLLOWED SOON THEREAFTER WITH A HDG CHANGE TO 290 DEGS. AT APPROX 3600-3800 FT WE RECEIVED AN RA TO DSND BY THE TCASII SHOWING WE NEEDED A DSCNT OF 2000 FT DOWN. THE TARGET WAS CLOSE AHEAD AND HIGHER; ALT NOT OBSERVED BY ME OR MY CREW. WE INITIATED A SMOOTH NOSE-OVER AND MAY HAVE REACHED 4300 FT OR MORE; BEFORE DSCNT INPUT TOOK EFFECT. I SAW A SOLID RED SQUARE AT THE 0130 O'CLOCK POS AND THIS WAS CONFIRMED BY THE FO; DISTANCE APPROX 1 MI. I BELIEVE I SAW PLUS 01 ON THE TCASII SCREEN. WE RECEIVED A TOTAL OF 2 RA COMMANDS TO DSND. I ESTIMATE WE HAD ABOUT A 1000 FT RATE OF DSCNT AND INCREASING WHEN THE RA CEASED AND WE WERE GIVEN THE CLR OF TFC ANNOUNCEMENT. WE RESUMED A CLB. THE CTLR ASKED ABOUT OUR DSCNT AND I ADVISED WE HAD FOLLOWED AN RA AND ASKED HIM IF HE SHOWED ANY TFC THAT WOULD HAVE CAUSED THE RA. HE ANSWERED THAT HE DID NOT HAVE ANY CONFLICTING TFC (NOT HIS EXACT WORDS). WE WERE GIVEN ADDITIONAL ALT CHANGES TO 15000 FT; 12000 FT; AND 9000 FT VERY QUICKLY AND ADDITIONAL HDG CHANGES AS WELL. VERY SHORTLY AFTER MY FIRST 'INTERROGATIVE' AS TO ADDITIONAL TFC THAT WOULD HAVE CAUSED THE RA; I ASKED AGAIN AND THE CTLR REPEATED HIS NEGATIVE REPLY. AT THAT TIME; ACR Y CAME ON THE FREQ AND SAID THAT HE HAD SEEN US; THAT HE HAD RECEIVED AN RA AT THE SAME TIME AND REQUESTED THE CTLR'S TELEPHONE NUMBER. I DID NOT TALK TO ACR Y MYSELF AS THE FREQ WAS VERY BUSY AND I DID NOT SEE THE NEED TO DEBATE THE ISSUE ON THE AIR. I DO NOT KNOW WHETHER ACR Y SAW MY ACFT VISUALLY OR ON THE TCASII SCREEN. I ALSO DO NOT KNOW WHETHER FLT ACR Y EXECUTED ANY RA MANEUVER.