He appears stumped. A rare enough occasion.

The topic that gives Brian Burke pause on this night?

No, not the brawl between his Calgary Flames and the Vancouver Canucks. (That particular icebreaker of a subject, by the way, is a non-starter in the visiting team’s bunker at Rogers Centre.)

What renders the Flames boss speechless during Saturday’s second intermission is a question about untouchables — as in, how many are on his roster?

Remember — heading into the off-season last spring, then-general manager Jay Feaster had singled out Sven Baertschi, T.J. Brodie, Curtis Glencross and Mark Giordano for hands-off status.

And now with Burke in charge?

“I haven’t thought about it,” he said after a drawn-out stretch of silence. “I’d have to give that more thought. I think when you’re where we are in the standings, it should be a pretty short list. There are untouchables.

“Give me a couple days to think about that.”

Well, for starters, here are three keepers — the Flames’ first-, second-, third-round selections.

Burke, with his squad in the midst of a teardown, insisted that he’s not swapping any of those draft choices.

“They are not in play,” he said. “We have to re-stock the cupboard.”

On the ice — and in the front office.

As Burke contemplates the reshaping of the Flames’ personnel, he has two vacancies — general manager and assistant manager — to soon fill. Even though insiders are thoughtful enough to provide him with his own short list — claiming that Burke’s first choice is Joe Nieuwendyk, while his runner-up is George McPhee, whose contract with the Washington Capitals expires this summer — he declined to offer even a hint about the hunt’s progress.

“We’re going to do it all privately and quietly,” said Burke, “then we’re going to have an announcement. So I have no comment on the search. We’re equipped to handle the (March 5 trade) deadline, even if we don’t hire a GM. I’m not worried about the (scouting) coverage.”

Nevertheless, the city awaits anxiously to see what the man has up his cuff-linked sleeve.