LR60 CREW HAS A CLOSE ENCOUNTER WITH A KING AIR ON CLBOUT FROM PSP.

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WE DEPARTED PSP ARPT AT XA23Z ON AN IFR FLT PLAN. IT WAS AN FAR PART 135 ON-DEMAND CHARTER FLT WITH 2 CREW AND 2 PAX. THE WX WAS VFR WITH UNLIMITED VISIBILITY AND CLR SKIES. WE WERE TOLD TO DEPART RWY HDG AND MAINTAIN 10000 FT. SHORTLY AFTER DEP; WE WERE GIVEN A TURN OF MORE THAN 90 DEGS FROM THE RWY HDG AND TOLD TO MAINTAIN 10000 FT. PSP TWR CTLR HANDED US OFF TO THE PALM SPRINGS DEP CTLR (FREQ UNKNOWN). WE CHKED IN AND HE ASKED US TO MAINTAIN THE PREVIOUSLY ASSIGNED HDG AND GAVE US A CLB TO 14000 FT. WE ACKNOWLEDGED THE INSTRUCTIONS AND WERE THEN GIVEN A TA FROM ATC. THE TFC WAS LOCATED AT 11-12 O'CLOCK POS; LESS THAN 5 MI AWAY AND WAS ABOVE OUR ALT. WE RPTED THAT THE TFC WAS NOT IN SIGHT; BUT WE WOULD BE LOOKING VISUALLY FOR THE KING AIR. ATC ADVISED THE KING AIR OF OUR POS; ALT AND CLB CLRNC TO 14000 FT. THE KING AIR ADVISED ATC THAT HE WOULD LOOK FOR US. WITHIN SECONDS OF THAT XMISSION; THE KING AIR CALLED ATC AND ADVISED THEM THAT HE HAD OUR ACFT IN SIGHT VISUALLY. I THINK ATC INSTRUCTED THE KING AIR TO MAINTAIN VISUAL SEPARATION FROM OUR ACFT; BUT I AM NOT SURE OF THIS. THE KNOWN FACT IS THAT WE HAD NOT ACQUIRED THE KING AIR VISUALLY AT THIS POINT. OUR CLB RATE WAS APPROX 4500 FPM DUE TO THE MOUNTAINOUS TERRAIN SURROUNDING PSP ARPT. SHORTLY AFTER THE KING AIR CALLED US IN SIGHT; WE RECEIVED A TCAS TA. WE CONTINUED THE CLB AND TURN AS PREVIOUSLY INSTRUCTED. WITHIN A COUPLE SECONDS; WE RECEIVED A TCAS RA THAT STATED 'ADJUST VERT SPD. DSND! DSND!' AT THIS POINT; WE WERE IN A R-HAND TURN TO THE ATC ASSIGNED HDG AND IN A STEEP CLB. OUR ALT WAS 7500 FT MSL AND THE KING AIR TCAS TARGET WAS SHOWING 7800 FT MSL. I IMMEDIATELY INCREASED OUR BANK ANGLE TO THE R (AWAY FROM THE KING AIR) AND REDUCED PWR. THE PWR REDUCTION AND INCREASED BANK ANGLE SIGNIFICANTLY REDUCED OUR CLB RATE; BUT WE STILL PASSED THROUGH THE KING AIR'S ALT BEFORE THE CLB STOPPED (DUE TO INERTIA). AT THE SAME TIME THAT WE RECEIVED THE TCAS RA; THE PIC IN THE L SEAT VISUALLY ACQUIRED THE KING AIR AT OUR 9-10 O'CLOCK POS AND SLIGHTLY ABOVE US. HE IMMEDIATELY ADVISED ME THAT HE HAD THE KING AIR IN SIGHT AND THAT WE WOULD MISS THE TARGET AND AVOID THE CONFLICT. HE ESTIMATED THAT WE PASSED WITHIN 500-1000 FT LATERALLY AND WITHIN 200 FT VERTICALLY. NO DIALOGUE OCCURRED BTWN OUR ACFT; THE KING AIR; OR ATC FROM THIS POINT FORWARD REGARDING WHAT HAD HAPPENED. I DOUBT THE KING AIR SAW US PASS UNDERNEATH HIM. WITHIN MINS OF THIS; PALM SPRINGS DEP CTL PASSED US TO ZLA WITHOUT ANY MENTION OF THE KING AIR OR THE CONFLICT. I THINK THERE WAS A BREAKDOWN OF COM BTWN ATC AND THE 2 ACFT. THE KING AIR MAY HAVE THOUGHT HE SAW US AND REALLY DID NOT; OR MAYBE HE SAW US AND THEN WENT HEADS DOWN IN THE FLT DECK. EITHER WAY; THIS EVENT COULD HAVE EASILY BEEN AVOIDED WITH BETTER ATC SEPARATION OR MORE DILIGENCE ON THE PART OF ALL PLTS INVOLVED.