UNDER ATC CTL; DSCNT CLRNC GIVEN WHICH WOULD HAVE DSNDED RPTR ACFT INTO ANOTHER LOWER ACFT. TCASII SHOWED THE TFC; RPTR QUESTIONED ATC.

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ATC FACILITY; GUAYAQUIL CTL. LOCATION; ESV VOR; SBOUND ALONG UG437; 5 DME ON THE 182 DEG RADIAL. WHEN APPROX 5 DME S OF ESV VOR; WE MADE OUR POS RPT TO GUAYAQUIL CTL; WHILE SIMULTANEOUSLY REQUESTING A DSCNT TO A LOWER ALT IN ORDER TO PREPARE FOR LNDG AT SEGU (GUAYAQUIL). ATC CLRED US TO DSND TO 9000 FT. WE BEGAN OUR DSCNT; AND WHEN PASSING FL350; WE NOTICED A TCASII TARGET DIRECTLY BELOW US; SAME DIRECTION OF FLT. I SWITCHED THE TCASII RANGE TO 5 MI; AND OBSERVED THAT THE TARGET WAS STATIONARY; APPROX 1/4 MI AT OUR 1 O'CLOCK POS; AND 2000 FT BELOW. I ELECTED TO HALT MY DSCNT TO FL340 TO AVOID A COLLISION. THE FO ADVISED GUAYAQUIL CTL OF THE SIT; AND THEIR RESPONSE WAS ONE OF TOTAL CONFUSION; AS THEY DID NOT HAVE A CLUE WHAT WE WERE TALKING ABOUT. I ELECTED TO SLOW TO MY MINIMUM MACH TO OBTAIN SPACING; AND WHEN 5 MI IN TRAIL OF THE TFC; AGAIN REQUESTED DSCNT WHEN I OBSERVED THE TFC ALSO IN A DSCNT. WE WERE CLRED TO FL310; AND WERE GIVEN NUMEROUS STEP DSCNTS ALL THE WAY TO LNDG AT GUAYAQUIL. IN ADDITION; WE WERE TURNED AWAY FROM THE LOC TO RWY 21 TO OBTAIN ADDITIONAL SPACING FROM THE TFC. AFTER LNDG; I OBTAINED THE FOLLOWING INFO FROM GND CTL; AND PERSONAL OBSERVATION. THE ACFT WAS AN AIRBUS A320. AN INFLT COLLISION WAS AVERTED DUE TO AND ONLY BECAUSE OF OUR REFUSAL TO DSND AS CLRED. SEPARATION WAS OBTAINED ONLY BECAUSE OF OUR ACTION; AND NOT BECAUSE OF ANY ACTION ON THE PART OF ATC.