Updated at 1pm

The European Union agreed on Friday to move talks forward with London on Britain's exit from the bloc as leaders warmed to the British premier, but there was little clarity on trade issues and Austria warned the Irish border issue remained a "riddle".

On the second day of a summit in Brussels, leaders said that "sufficient progress" was made after a deal on citizens' rights, the Irish border and Britain's outstanding payments, giving negotiators a mandate to move onto the main phase of talks.

"EU leaders agree to move on to the second phase of Brexit talks. Congratulations PM Theresa May," European Council President Donald Tusk, who chairs EU summits, said on Twitter.

Leaders said that "sufficient progress" was made after a deal on citizens' rights, the Irish border and Britain's outstanding payments

Lithuanian President Dalia Grybauskaite said those talks would likely formally start in March.

A day after she suffered a defeat in parliament over her blueprint for quitting the EU, May won applause from her peers on Thursday evening. She said she was on course to deliver Brexit and urged them to speed up the talks to unravel more than 40 years of membership.

But several hours later as leaders reconvened, European Commission President Jean-Claude Juncker and Italy's Prime Minister Paolo Gentiloni warned that the hardest decisions were still to come as Britain tries to extricate itself from rules agreed over years in the bloc.

Austrian Chancellor Christian Kern went further, saying even a primary school student could see that the "first phase" deal on the Irish border would come back to haunt the talks because it was impossible for Britain to leave the bloc's single market while avoiding a hard border on the island of Ireland.

"There cannot be any border controls between Northern and southern Ireland, there cannot be border controls between Northern Ireland and the UK, but there can between UK and the EU," he said.

"So our primary school students can see that there is a riddle to be solved."

As May left to return to London - she did not join the other 27 leaders for further discussions on Brexit and the eurozone - she said she was eager to move on.

The next, more decisive phase of the negotiations will further test her authority by exposing the deep rifts among her top team of ministers over what Britain should become after Brexit

The EU is willing to start talks next month on a roughly two-year transition period to ease Britain out after March 2019, but has asked for more detail from London on what it wants before it will open trade negotiations from March of next year.

Ireland's Prime Minister Leo Varadkar cautioned that there were "quite divergent opinions" on how a new relationship and transition would look. EU officials are divided over whether Britain should continue to receive the full, unfettered economic benefits of EU membership during a transition after it leaves, even if it loses political representation in Brussels.

A British government official said the prime minister was approaching the next phase, which will discuss a transition period as well as the terms of the future trading relationship, "with ambition and creativity".

German Chancellor Angela Merkel gave her stamp of approval, but cautioned time was running out.

"We made clear that Theresa May has made an offer that should allow us to say that we have seen sufficient progress," she told reporters. "Nevertheless, there are still a lot of problems to solve. And time is of the essence."

Tusk will call May on Friday to update her.

May, weakened after losing her Conservative Party's majority in a June election, has so far carried her divided government and party with her as she negotiated the first phase of talks on how much Britain should pay to leave the EU, the border with Ireland and the status of EU citizens in Britain.

But the next, more decisive phase of the negotiations will further test her authority by exposing the deep rifts among her top team of ministers over what Britain should become after Brexit.