ACR L1011 AND PIPER SARATOGA HAVE CLOSE ENCOUNTER CLOSE TO PBI.

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WE HAD DSNDED INTO THE TFC PATTERN AT PBI. OUR FIRST APCH CTLR CLRED US TO DSND TO 4000 FT ON A 180 DEG HDG. SHORTLY THEREAFTER; WE WERE HANDED OFF TO THE FINAL APCH CTLR. AFTER OUR INITIAL CONTACT WITH THE FINAL APCH CTLR; WE WERE GIVEN AND COMMENCED A DSCNT TO 3000 FT. WE ALSO BEGAN A SPD REDUCTION FROM 250 KIAS TO 210 KIAS TO INITIATE FLAP EXTENSION. BEFORE WE REACHED 3000 FT; THE FINAL CTLR GAVE US A DSCNT TO 2000 FT AND A TURN TO 095 DEGS. HE SIMULTANEOUSLY GAVE US A TA; 'ACFT X CHK TFC; A SARATOGA (Y) AT 11 O'CLOCK POS.' I DON'T REMEMBER THE ALT OF THE TFC AS WE WERE IN THE MIDST OF CONFIGURING THE ACFT FOR THE APCH. INCIDENTALLY; THERE EXISTED A BROKEN CLOUD LAYER BTWN 2500 FT AND 4000 FT OVER THE BEACH AREA PARTIALLY OBSCURING PBI AND THE GND. VISIBILITY BENEATH THIS CLOUD LAYER WAS 10+ MI. OUR DSCNT RATE WAS SHALLOW AND I BELIEVE THAT WE WERE DSNDING THROUGH 3500 FT AT THIS TIME. THE FO WAS FLYING AND I WAS HANDLING THE RADIOS. THE SO AND OUR JUMP SEAT RIDER WERE LOOKING OUTSIDE AND ACCOMPLISHING OTHER NECESSARY TASKS. AS WE COMMENCED FURTHER CONFIGN FROM FLAPS 4 DEGS TO FLAPS 10 DEGS; WE NOTED SEVERAL HOLLOW WHITE DIAMONDS ON THE TCASII SCOPE WHICH WAS ON THE 5 MI RANGE. I ATTRIBUTED THIS TO OUR LOCATION OVER LANTANA ARPT AND CLOSE PROX TO WEST PALM BEACH INTL ARPT. WE SAW NO TFC. ALL OF A SUDDEN IN OUR 11-12 O'CLOCK POS; ONE OF THE DIAMONDS WENT FROM HOLLOW TO FILLED WHITE; THEN TO FILLED YELLOW. WE THEN RECEIVED 'TFC; TFC' ADVISORY. WHEN THE YELLOW DIAMOND BECAME A RED SQUARE; IT WAS WITHIN THE DOTTED CIRCLE (LESS THAN 2 MI AWAY) AND THE RA COMMANDED AN IMMEDIATE CLB. WE NOTED EITHER A 02 OR 05 BENEATH THE RED SQUARE. AS THE FO INITIATED THE EVASIVE MANEUVER (CLB); WE RECEIVED A 'REVERSAL RA' WHICH STATED 'DSND NOW.' BOTH THE FO AND I PUSHED FORWARD ON THE YOKE TO INITIATE THE DSCNT. AT THIS POINT; THE JUMP SEAT RIDER AND I PICKED UP THE TFC. IT APPEARED AS THOUGH IT WAS COMING OUT OF THE CLOUDS AT US IN OUR 12 O'CLOCK POS. IT WAS CLOSE; REAL CLOSE. THE CTLR ASKED US IF WE WERE READY FOR A TURN TO FINAL AND I DECLINED STATING THAT WE NEEDED TIME TO REGROUP AFTER THIS INCIDENT. THE CTLR GAVE US A 180 DEG HDG AND 2000 FT. AFTER 2-3 MINS; WE REQUESTED VECTORS FOR THE APCH. WE LANDED; UNEVENTFULLY; ON RWY 31 AT PBI. AFTER LNDG; I CONTACTED THE SHIFT SUPVR AT PALM BEACH APCH CTL VIA LAND LINE. HE INFORMED ME THAT HE RELIEVED THE CTLR AND WAS ABOUT TO REVIEW THE TAPES. I TOLD HIM THAT I DID NOT WANT THE CTLR PUT INTO JEOPARDY NOR DID I WISH HIM ADMONISHED. HOWEVER; I WAS INTERESTED IN WHY A LIGHT PVT ACFT WAS PERMITTED TO FLY VFR IN AND OUT OF THE CLOUDS; AT NIGHT; UNLIT WITHIN CLOSE PROX OF TWO RELATIVELY BUSY ARPTS.