LGT ON INITIAL CLB HAS TCASII ALERT AND NMAC WITH LIGHT TWIN.

Narrative

THE FLT DEPARTED RWY 27R AT FLL ENRTE TO THE NEW YORK AREA (JFK). THE TKOF WAS NORMAL. AT APPROX 800 FT AGL WE WERE ALERTED TO TFC BY TCASII ('TFC; TFC'). I LOOKED AT THE TCASII DISPLAY TO DETERMINE THE LOCATION OF THE CONFLICTING TFC. THE TFC WAS AT OUR 11 O'CLOCK AND 2 MI AND 300-400 FT ABOVE OUR ALT AT THAT TIME. THE TFC WAS SLOW MOVING AND WE COULD NOT IMMEDIATELY DETERMINE ITS DIRECTION OF FLT. WE CONTINUED THE CLB AND MADE A SHALLOW R TURN AS IT BECAME APPARENT THAT THE TFC WAS HDG W TO NW. THE TFC WHICH WAS NOW IDENTIFIABLE AS A LIGHT TWIN BEGAN A R TURN TO THE N WHILE CLBING. HE WAS OBVIOUSLY NOT AWARE OF OUR PRESENCE. BECAUSE THE TFC WAS STILL SLIGHTLY ABOVE US AND ABOUT TO CROSS OUR FLT PATH IN ITS R TURN WE ELECTED TO CHANGE OUR COURSE OF ACTION. WE ALSO BEGAN A TURN TO THE N AND BEGAN AN IMMEDIATE DSCNT TO INCREASE THE VERT SEPARATION. THE TFC PASSED ALMOST DIRECTLY OVERHEAD WITHIN 200-400 FT. AFTER DSNDING TO APPROX 800 FT AGL WE CONTINUED THE CLB TO OUR ASSIGNED ALT. FLL TWR POINTED OUT THE TFC AFTER WE HAD ALREADY BEGUN TO MANEUVER AROUND IT. AFTER PASSING THE TFC WE WERE ASKED BY THE TWR IF WE HAD VISUAL CONTACT WITH TFC. WE WERE TOLD TO RETURN TO A 270 DEG HDG IN A MANNER WHICH LED US TO BELIEVE THAT THE TWR HAD NO IDEA WHY WE HAD MADE THE TURN TO THE N. THEY DIDN'T SEEM TO KNOW ANYTHING ABOUT THE TFC INCLUDING FROM WHENCE IT CAME. THE FLT TO JFK THEREAFTER WAS UNEVENTFUL.