BRUSSELS — British Prime Minister Theresa May made a whirlwind tour Tuesday of European capitals to try to win further concessions on her deal to pull Britain from the European Union, a day after her plan’s unpopularity led her to cancel a key parliamentary vote.

But the issue that has made the Brexit deal politically toxic in Britain is nonnegotiable on the European side, and it was unclear whether May could gain anything to improve her political math as she bounced from The Hague to Berlin to Brussels. 

Reflecting rising European fears that Britain could crash out of the E.U. without a deal, the chief E.U. Brexit negotiator, Michel Barnier, urged colleagues to speed their preparations for the humanitarian and economic crisis that would ensue.

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Leaders offered kind words to May and said they would do everything they could to help her sell the deal to the British Parliament, but they refused to back down from an ironclad backup plan to ensure there will never be a hard border between the Republic of Ireland and Northern Ireland.

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“There is no room whatsoever for renegotiation,” European Commission President Jean-Claude Juncker told the European Parliament in Strasbourg on Tuesday. “Ireland will never be left alone.”

Declaring himself “astonished” at May’s inability to master her domestic politics, Juncker said that Europeans could give “further clarifications and further interpretations” to help ease the British debate — diplomatic code for declarations that could be politically symbolic but ultimately will not be substantive or binding.

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E.U. leaders plan to discuss Brexit on Thursday during a previously scheduled summit in Brussels.

European ministers who gathered ahead of that meeting said, one after another, that they could imagine no change in the fundamentals of the deal.

“We have done a lot to help the U.K. in its withdrawal agreement,” French Minister for European Affairs Nathalie Loiseau said. “It is the only possible agreement. And we have done a lot of concessions to reach it.”

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She said France had begun to brace for Britain’s leaving the E.U. in March without any deal, a situation that could halt travel and trade. France will “take all necessary measures” to prepare for a crisis, she said.

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German officials offered similarly no-nonsense warnings. Asked whether the E.U. could do anything to help May, German Minister of State for Europe Michael Roth said “it’s always good to talk to each other.”

But on Tuesday, leaders appeared to be talking past each other. Roth’s British counterpart, Martin Callanan, said May wanted “additional, legal reassurances that the U.K. cannot be permanently trapped in the Irish backstop,” using the term for the backup plan that would keep an open border on the island of Ireland. “That’s been the issue all along, and that’s the issue at the heart of the concerns expressed by many members of Parliament.”

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By the end of the day, the limit of May’s negotiating position was apparent.

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“There is a shared determination to deal with this issue and address this problem,” she said after finishing her meetings. She repeated the line she had used going in: The current agreement is “the best deal available. Indeed, it’s the only deal available.”

As she traveled across Europe, May received friendly receptions, even if leaders offered nothing to alter Britain’s domestic political calculus.

The polite discussions contrasted with her appearance in Britain’s House of Commons on Monday. When May announced that she was delaying the vote on her Brexit deal, that she had “listened very carefully to what’s been said in this chamber and out of it,” she was jeered.

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May decided she had little choice but to put off the vote after it became clear she could face more than 100 defections from her own Tory party.

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British lawmakers, who foresee little progress in more E.U. talks, decried May’s decision to postpone the vote. A Downing Street spokesman said Tuesday that a vote would be held “before January 21” — which leaves flexibility ahead of Britain’s March 29 exit date.

In London, the ticking clock has angered both those who want to leave the E.U. and those who wish to remain.

“We cannot afford to waste the next six weeks in talks going nowhere,” Conservative Party lawmaker Damian Collins said.

Some political analysts say May is so seriously wounded that she may not be able to hang on to office much longer — a dynamic that could make it even harder to win concessions from Brussels.

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“She’s completely lost the authority of the Commons. You can’t go on like that,” said Jonathan Tonge, a politics professor at the University of Liver­pool. 

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“Unless she can produce some Brussels rabbit out of the hat,” he said, it’s hard to “see where she can go from here politically.”

In dispute is a provision that would go into effect if Britain and the E.U. fail to find a better deal during a transition period set to last until the end of 2020. 

The Republic of Ireland, which is remaining part of the E.U., wants to ensure that its border with departing Northern Ireland stays open to prevent a recurrence of the violence that plagued the island for decades. Free movement and free trade help tamp down tensions by allowing Northern Irish republicans, who resist British authority, to feel more connected to the Republic of Ireland.

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London, in turn, refuses to countenance any plan that would split apart Northern Ireland and the rest of the United Kingdom — thus forcing the whole country to remain inside the E.U. customs union, with little freedom to negotiate independent trade deals.

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The tensions were apparent in London on Tuesday, where a shouting protester was subdued with a stun gun at the gates of Parliament, according to British media reports.

In Berlin, when May arrived for lunch with German Chancellor Angela Merkel, the British leader got stuck in the back of her armored Mercedes, as aides struggled to unlock the door. It was unclear whether the child safety lock was the cause.

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The incident led some British commentators to recall the warning from Environment Secretary Michael Gove, who campaigned for Brexit ahead of the 2016 referendum by warning that “if we vote to stay, we’re not settling for a secure status quo. We’re voting to be hostages locked in the back of the car and driven headlong towards deeper E.U. integration.”

Booth reported from London. Karla Adam in London and Quentin Ariès in Brussels contributed to this report.

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