MD80 FLC INITIATE TCASII RA DSCNT EVASIVE ACTION; AND EVASIVE MANEUVER IS CHALLENGED BY APCH CTLR. PLT CONCERNED ABOUT RECEIVING CONFLICTING INFO; TCASII AND ATC.

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INBOUND TO LGA ON MINKS 1 ARR; 12000 FT; RECEIVED TFC NOTIFICATION FROM APCH FREQ 125.45 -- COULD NOT SEE TFC. FINALLY RECEIVED A DSNDING RA. NOTIFIED APCH CTL THAT WE WERE RESPONDING TO AN RA; BUT CTLR'S RESPONSE WAS 'DO NOT DSND.' WE FOLLOWED RA TO ABOUT 250 FT BELOW ASSIGNED ALT; WHEN TCASII GAVE A 'MONITOR VERT SPD.' WHEN THAT CEASED; WE RETURNED TO ASSIGNED ALT. WHAT CONCERNS ME IS THE CTLR'S ISSUING INSTRUCTIONS TO IGNORE TCASII. WE COULD NOT FIND THE TFC; WE RESPONDED TO AN RA; WE NOTIFIED ATC; AND RETURNED TO ALT ASAP. THESE ARE THE RULES AS I UNDERSTAND THEM. WHAT WAS OBVIOUSLY CONFUSING WAS RECEIVING CONFLICTING INSTRUCTIONS FROM TCASII AND ATC. I DON'T BELIEVE ATC SHOULD BE GIVING INSTRUCTIONS TO IGNORE AN RA. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR VERIFIED THAT HE WAS IN N90 AIRSPACE; IN THE IMMEDIATE VICINITY OF MINKS INTXN; CLRED DIRECT LGA. THE PLT DID NOT SEE EITHER ACFT ABOVE OR BELOW. THE ACFT ABOVE; HE THOUGHT; CAME WITHIN 600 FT VERT; POSSIBLY A VFR ACFT. ONLY 1 TIME BEFORE HAD A CTLR ASKED HIM NOT TO REACT TO TCASII; BUT THAT WAS RIGHT AFTER THE TCASII SYS WAS BROUGHT INTO SVC. THE PLT WAS AWARE OF THE 'BOX' A TCASII RA CAN PUT THE CTLR INTO -- SEEING SOMETHING HAPPEN; BUT NOT BEING ABLE TO DO MUCH ABOUT IT.