British MPs have given their support to Theresa May returning to Brussels to attempt to renegotiate the controversial Northern Irish backstop, though with the EU firmly insisting the withdrawal agreement is not open to negotiation, the path ahead remains unclear. The Labour Party has also agreed to hold talks with the government over Brexit after MPs supported a non-binding resolution that opposes Britain leaving the EU without a deal.

The developments came in a series of amendments to May’s withdrawal plans on Tuesday evening as MPs have sought to break the political impasse over the direction of Brexit, after voting by a substantial majority earlier this month to reject her deal as it stood.

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The key victory for the government came in the support for an amendment by Sir Graham Brady, who heads the Conservative Party’s 1922 backbench committee. The amendment calls for an alternative to the Irish backstop or insurance scheme that has been at the heart of some Conservative and all Democratic Unionist Party opposition to the deal. The backbench amendment, which calls on the government to return to the negotiating table with the EU and exact substantive changes, was backed by the government in the end and won by 317 to 301 following an afternoon of heated debate.

While May spun it as a victory – she insisted that it was “clear that there is a route that can secure a substantial and sustainable majority” for leaving with a deal – it remains highly uncertain what concessions Britain can get and whether they will be enough to satisfy MPs when the deal is voted on again in coming weeks. Should no substantial changes be obtained, the MPs who on Tuesday gave May the benefit of the doubt, could return to opposing the deal as they did in the initial vote on the withdrawal agreement earlier this month.

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Following the vote, the European Council President Donald Tusk insisted that the withdrawal agreement “is and remains the best and only way to ensure an orderly withdrawal of the United Kingdom,” and that the backstop and withdrawal agreement were not open for renegotiation. A similar response came from the Irish government. However, hard Brexit supporters remained adamant that change was possible. When it came to the EU, amending the backstop was “no skin off their nose,” former Foreign Secretary and arch Brexiteer Boris Johnson told the BBC.

While the government won on the crucial Brady amendment it suffered a significant defeat when an amendment by Conservative MP Caroline Spelman was passed by the house, stating that Britain would not leave without a deal. While the vote is non-binding it is seen as an indication of Parliament’s determination to act in the future to prevent such a scenario unfolding. However, it does not take no-deal off the table. An amendment by Labour MP Yvette Cooper that would have delayed Brexit by three months if no plan was agreed by the end of February – effectively providing the means for MPs to rule out no deal – was defeated.