A TFC ALERT IN THE CLASS A AIRSPACE GIVEN BY ARTCC AND FOLLOWED BY THE CREW. EVASIVE ACTION WAS CONTRARY TO THE TCASII RA. NO LOSS OF SEPARATION OCCURRED. THE SECOND ACFT HAD NO TCASII INSTALLATION.

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PRIOR TO THIS INCIDENT AND THROUGHOUT THE FLT; ALL CHANGES IN HDG AND ALT WERE REQUESTED FROM ATC AND CLRNC WAS OBTAINED PRIOR TO BEING INITIATED. A REQUEST WAS MADE TO ATC FOR PERMISSION TO CLB TO FL350 AND TO REVERSE COURSE. CLRNC WAS RECEIVED AND ACKNOWLEDGED TO CLB TO FL330 AND TO TURN TO 230 DEG HDG; WITH CLRNC TO FL350 AND THE DESIRED S HDG TO COME LATER. A L TURN AND CLB WAS BEGUN. IN A CLBING L TURN; APCHING FL330; A CALL WAS RECEIVED FROM CTR TO LEVEL AT FL310 IMMEDIATELY; FOLLOWED SHORTLY BY A MORE AGITATED CALL; SPECIFYING 'TFC ALERT;' DSND TO FL310 IMMEDIATELY; AND REPEATED AT LEAST ONCE. THE AUTOPLT WAS DISCONNECTED AND A PUSHOVER WAS INITIATED AS THE L TURN WAS CONTINUED AND THE ACFT DSNDED TO AND LEVELED AT FL310. ALMOST SIMULTANEOUSLY WITH THE 'TFC ALERT' VOICE CALLS; THE TCASII ISSUED A RA; TURNING THE INST FACE RED; AND XMITTING THE WORDS 'TFC; TFC; CLB; CLB' ON THE INTERCOM. THE CONFLICTING ACFT WAS AT ABOUT 2 O'CLOCK POS RELATIVE BEARING. IT WAS NEVER VISIBLE TO THE PF IN THE L SEAT. A BRISK PUSHOVER WAS ALREADY IN PROGRESS; IN ACCORDANCE WITH THE ATC COMMAND; BUT OPPOSITE TO THE TCASII SUGGESTION. THE PF ELECTED TO CONTINUE PUSHING. ITS LIGHTS WERE SEEN BY THE PNF IN THE R SEAT; WHO COULD NOT DISCERN IF IT WAS HIGHER OR LOWER; BUT HE DID DETERMINE THAT IT WAS MOVING AFT RELATIVE TO US AND WOULD PASS BEHIND. IN THE ENSUING CONVERSATION; IT WAS OBVIOUS THAT THE CTR CTLR WAS CONCERNED THAT THE CONFLICT HAD EXCEEDED SAFE SEPARATION STANDARDS. HE ASKED HOW CLOSE TO FL330 WE HAVE GOTTEN; BUT WE WERE NOT ABLE TO SAY PRECISELY. LATER; HE REQUESTED THAT WE CALL THE CTR AFTER LNDG. AFTER LNDG; A PRINTOUT OF TRAJECTORY DATA WAS OBTAINED; WHICH SHOWED THAT WE HAD JUST GRAZED FL330 DURING THE PUSHOVER MANEUVER. WE DID CALL THE WATCH SUPVR. HE THANKED US FOR RESPONDING; AND SAID THAT THEY HAD DETERMINED THAT A SAFE SEPARATION VIOLATION HAD NOT OCCURRED; AND THAT THEY CONSIDERED THE INCIDENT CLOSED. HE MENTIONED THAT THE OTHER ACFT WAS A FLT NOT EQUIPPED WITH TCASII. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR WAS NEW TO THE USE OF TCASII THAT WAS RECENTLY INSTALLED IN HIS ACFT. HE THOUGHT THAT IF HE FLEW THE RA THE POTENTIAL CONFLICT WOULD HAVE BEEN CLOSER BUT A SAFE RESOLUTION JUST THE SAME.