ACR GIVEN CONFUSING CLRNC WITH SECOND ACFT ON DSCNT INTO LGA.

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NEW YORK CENTER CLRED OUR FLT FROM FL280 TO 10000 FT AND 250 KTS AT VALRE INTXN. WE HAD NOT BEEN CLRED FOR THE EXTOL 1 ARR. PASSING 12000 FT CENTER CALLED SLOW MOVING TFC AT 12 O'CLOCK; 10500 FT; 5 MI. AT 10800 FT AND DSNDING CENTER TOLD US TO STOP DSCNT AT 11000 FT (APPARENTLY PREVIOUSLY CALLED TFC WOULD NOT BE CLR OF OUR FLT PATH). WE NOW QUESTIONED THE CTLR WHETHER HE WANTED US TO CLB BACK UP TO 11000 FT OR CONTINUE ON ORIGINAL CLRNC TO 10000 FT. WE WERE NOW AT 10500 FT WITH ALL EYES OUTSIDE LOOKING FOR A TARGET WE THOUGHT HAD CLRED OUT OF OUR PATH. CENTER'S REPLY WAS TO CONTINUE TO 10000 FT THE TFC WOULD PASS OFF OUR R; INTERCEPT J37 S OF IGN; AND CHANGE TO NEW YORK APCH CTL FREQ. NEW YORK APCH CTL INFORMED US WE HAD MISSED OUR TURN ONTO THE DPK 336 DEG RADIAL AND OUR SPD SHOULD BE 250 KTS; WHEN IN FACT THE PREVIOUS CTLR HAD CANCELLED THE 'NEVER GIVEN EXTOL 1 ARR' BY GIVING US 'INTERCEPT J37 S OF IGN'. CONTRIBUTING FACTORS: NEW YORK CENTER HAD APPARENTLY GOTTEN DISTR AND WAS LATE GIVING US A LEVEL OFF TO AVOID TFC. IN THE CONFUSION HE THEN CLRED US J37 RATHER THAN THE EXTOL 1 ARR AND DID NOT TELL NEW YORK APCH WHAT HE HAD DONE.