AN ACR LTT DIVED IN RESPONSE TO A TCASII COMMAND.

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WE WERE INSTRUCTED TO FLY A 190 DEG HDG AND JOIN THE LDA- A LOC TO RWY 22 AT LGA; WITH A DSCNT CLRNC TO 3000 FT. WE WERE APPROX 15 MI NE OF LGA AND 3 MI N OF THE LOC; DSNDING AT 1000 FPM AND 195 KTS; WHEN WE RECEIVED A TA FROM TCASII FOR A TARGET AT 10-11 O'CLOCK AND 3 MI; 300 FT BELOW OUR ALT. ONLY A FEW SECONDS LATER; OUR TCASII ISSUED A 'DSND; XING DSND' RA WITH A 1500 FPM COMMANDED DSCNT. OUR DSCNT RATE WAS INCREASED TO THE COMMANDED RATE AND I OBSERVED NO CHANGE IN OUR RELATIVE ALT WITH THE OTHER ACFT; STILL 300 FT BELOW US. A FEW MORE SECONDS PASSED AND OUR TCASII COMMANDED 'INCREASE DSCNT' AND COMMANDED A 2500 FPM DSCNT RATE. I OBSERVED THE TARGET'S RELATIVE ALT INCREASE AND VISUAL CONTACT WAS MADE AS THE TFC PASSED OUR 12 O'CLOCK POS AND APPROX 100 FT ABOVE OUR ALT; WITH THE TCASII INDICATING THE TFC PASSED 100 FT ABOVE US. I NOW LEVELED OFF AT OUR PRESENT ALT OF 2000 FT; STILL ON A 190 DEG HDG. I MAINTAINED THIS ALT AND HDG AWAITING FURTHER INSTRUCTIONS. WE WERE THEN INSTRUCTED TO FLY A 180 DEG HDG AND CLB TO 3000 FT. AFTER THE ENCOUNTER; MY FO INFORMED ME THAT HE HAD RESPONDED TO A HDG CHANGE OF 090 DEGS WHICH WAS GIVEN AT THE SAME TIME THAT OUR TCASII ISSUED THE 'INCREASE DSCNT' COMMAND. DUE TO THE VERY HIGH WORKLOAD; I MISSED THE RADIO CALL. IN RETROSPECT; I DO NOT BELIEVE THAT THE HDG CHANGE WOULD HAVE PROVIDED ANY RELIEF FROM THE CONFLICT AND; FROM MY POINT OF VIEW; WE WERE IN AN EMER SIT; WITH MY FIRST PRIORITY TO FLY THE ACFT TO SAFETY. I BELIEVE THIS CONFLICT WAS ATC CTLR ERROR. THE OTHER ACFT WAS NEVER POINTED OUT TO US BY THE CTLR AS TFC. THE OTHER ACFT WAS ALSO TCASII EQUIPPED THEREFORE; THIS WAS A COORDINATED MANEUVER. THE OTHER ACFT'S APPARENT DELAY IN RESPONDING TO THEIR OWN RA INCREASED THE SEVERITY OF THIS ENCOUNTER. SUPPLEMENTAL INFORMATION FROM ACN 254516: AT THE SAME TIME; A JET WAS CLRED FOR THE VISUAL. IT SEEMED AS IF THE JET WAS NOW DSNDING INTO US. THE TCASII WAS SO LOUD THAT THE CAPT DID NOT HEAR THE ATC HDG INSTRUCTIONS.