By Elizabeth Piper and Elisabeth O'Leary
LONDON, May 27 (Reuters) - Britain's two main parties promised on Monday to work to resolve the deadlock over Brexit, hoping to win back voters who abandoned them for a new movement led by eurosceptic Nigel Farage at European elections.
After a punishing night when Britain's deepening Brexit divisions were on show, the governing Conservatives said the results were a demand for the country to follow through on plans to exit the European Union no matter what.
Taking a different tack, the opposition Labour Party described a public vote - a new national election or second referendum - as a way to reunite the country.
With Farage's Brexit Party capturing the greatest number of votes for seats in the European Parliament, followed by a group of fervently pro-EU parties, Conservatives and Labour were under pressure to commit clearly to either side of the debate.
Almost three years since Britain voted narrowly to leave the EU and just under two months after the originally planned departure date, lawmakers are still arguing over how, when or even whether the country will leave the club it joined in 1973.
For the Conservatives, who will appoint a new leader by the end of July, many of the contenders will see the European vote outcome as proof that they must seek a cleaner break with the EU. For Labour leader Jeremy Corbyn, pressure will mount to embrace a second referendum without qualification.
But what is clear for both from a vote which many used as a protest, Brexit, which has already forced Prime Minister Theresa May to say she will resign on June 7 over her failure to deliver Brexit, risks shattering their election prospects.
Britain's interior minister Sajid Javid, one of several Conservatives who hope to replace May, described his party's performance in the election as "hugely disappointing".
"This is a verdict on our delivery of Brexit," Javid said on Twitter. "There's a clear lesson: people want us to get on with it."
Labour's finance chief, John McDonnell, also said it was time to unite but offered a different solution - a "public vote" to help pin down where most Britons stood on the way forward.
He caused a short-lived sensation in seeming to signal a shift in Labour policy to unequivocally back a second referendum on Britain's membership of the EU.
McDonnell later clarified that his, and his party's, first priority was to try to force a new election.
"Can't hide from hit we took last night. Bringing people together when there's such a divide was never going to be easy," he said on Twitter.
"Now we face prospect of Brexiteer extremist as Tory (Conservative) leader (replacing May) and threat of no deal, we must unite our party and country by taking issue back to people in a public vote."
After May announced on Friday that she was stepping down after failing three times to secure parliamentary approval of her EU withdrawal plan, many of her would-be successors have said they wanted Britain to leave the EU with or without a transition deal on the new exit date of Oct. 31.
The response seemed to be a direct challenge to Farage, a former commodities broker whose campaigning helped force May's predecessor, David Cameron, to stage the 2016 referendum.
After the Brexit Party came out tops with 31.6% of the vote, the 55-year-old Farage said on Monday he wanted to be included in any new negotiation to leave the EU.
"If we don't leave on Oct. 31 then the score that you have seen for the Brexit Party today will be repeated in a general election and we are getting ready for it," Farage said in Southampton, southern England.
But while the Brexit Party came first, three emphatically pro-EU parties - the Liberal Democrats, Greens and Change UK - also performed well, combining for more than 30% of the vote.
In total, England, Scotland, Wales and Northern Ireland elects 73 MEPs (members of the European Parliament) to the 751-seat assembly. They will not contribute directly to British policymaking on domestic issues like Brexit, but will have a say in EU-wide policy as long as they remain in the assembly.
Pro-Europeans fear Brexit will make them poorer, undermine London's position as a global financial capital and weaken the West as it grapples with Donald Trump's unpredictable U.S. presidency and growing assertiveness from Russia and China.
The Liberal Democrats, who campaigned under the slogan "Bollocks to Brexit", oppose Brexit and want a second referendum to stop it.
Supporters of Brexit admit there may be some short-term pain for the UK’s $2.9 trillion economy but say that long-term it will prosper when cut free from the EU, which they describe as a failing German-dominated experiment in European integration.
"Far from providing a clear verdict, the result simply underlined how difficult it is likely to be to find any outcome to the Brexit process that satisfies a clear majority of voters," said John Curtice, a leading polling expert.
(Reporting by Elizabeth Piper)
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