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UPDATE 1-Ball in your court, weakened May tells EU on Brexit talks

* May to try to advance Brexit negotiations

* Weakened leader faces party infighting

* EU negotiators see no imminent breakthrough

* But EU leaders might offer May a hand (Adds Jenkin comments, BAE Systems)

LONDON, Oct 9 (Reuters) - British Prime Minister Theresa May will tell the European Union on Monday the "ball is in their court" in Brexit talks, trying to advance negotiations despite a party conference that left her weaker than ever.

May's keynote speech at last week's conference, meant to revive her premiership, was marred by a repetitive cough, a prankster handing her a bogus employment termination notice and a stage malfunction. That came after May called a snap election in June and lost her party its majority in parliament.

She has so far fought off attempts to unseat her, and now wants to refocus on Brexit talks to unravel more than 40 years of union - with just 18 months to go before the country leaves the EU and must redefine its place in the world.

But she faces more disquiet in her Conservative Party over the negotiations and how a future relationship should look.

One lawmaker has urged her to "call time" on the talks if they fail to move forward at a summit this month.

With Brussels quietly preparing for a collapse in the talks and Britain getting ready for what May calls "all eventualities", some officials and business chiefs worry the country will crash out of the EU without a deal.

Speaking in parliament later on Monday, May will say she is determined to secure a new partnership with the other 27 members of the wealthy political and trade bloc.

"Achieving that partnership will require leadership and flexibility, not just from us but from our friends, the 27 nations of the EU," she will say, according to excerpts of her speech.

"And as we look forward to the next stage, the ball is in their court. But I am optimistic we will receive a positive response."

TRADING CONDITIONS

May, who will host businesses later on Monday to listen to their Brexit concerns, is keen to push the talks beyond a discussion of the divorce to try to offer firms some certainty about future trading conditions.

A report that aerospace manufacturer BAE Systems is planning to cut more than 1,000 jobs can only add pressure for May to press her case for the EU to move beyond talks on the divorce settlement, the rights of expatriates and the border with EU-member Ireland.

EU negotiators say that while they see no big breakthrough at the summit next week, they may offer May a hand by offering some hope of a shift at the next scheduled meeting in mid-December.

Aides to May have signalled that the prime minister has accepted that her October deadline will not be met despite a speech in Italy last month which attempted to reset the tone of the difficult negotiations.

But some pro-Brexit campaigners are calling on the prime minister to get ready to step away from the talks - underlining the deep divisions in the Conservative Party, which have again come to the fore, with lawmakers briefing against finance minister Philip Hammond, who supporters a so-called softer Brexit.

"We are fast reaching the point when the prime minister should assert the authority of her office over the negotiations and call time," Bernard Jenkin, a Conservative lawmaker, wrote in the Guardian newspaper.

Asked whether he meant Britain should walk away if there was no progress at the October summit, Jenkin told Reuters: "Yes". (Editing by Janet Lawrence)