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* Talks on citizens' rights, finance, legal issues, Ireland

* Aim is to agree divorce deal so free trade talks can begin

* Britain leaves EU on March 30, 2019 - with or without deal

By Alastair Macdonald

BRUSSELS, July 17 (Reuters) - Dozens of British officials and their EU counterparts got the first full round of Brexit negotiations under way on Monday, bringing a quiet buzz to the European Commission's Brussels headquarters.

After a brief handshake and sit-down for the cameras between Brexit Secretary David Davis and chief EU negotiator Michel Barnier, the pair left the real work of the week to teams of civil servants divided into groups by topic.

Davis returned to London but will be back on Thursday, officials said, in the hope that he and Barnier can give their political endorsement to a first set of partial agreements.

Barnier's team is known as the Article 50 Task Force after the provision of the EU treaty under which Britain has opted to quit the European Union. Numbering about 40, most are involved in the talks in the EU executive's Berlaymont headquarters.

Officials described a bustle as negotiators and staff moved among meeting rooms scattered along carpeted corridors and up and down lifts over nearly half of the cross-shaped building's 13 floors.

Barnier, a former French cabinet minister who much prefers to speak his native tongue, greeted Davis in English, the only language most British civil servants can work in. Though Barnier insisted French be one of the two languages for negotiations, English is the lingua franca of Brussels and will predominate.

NO NOTES

The meeting between Davis and Barnier featured just two aides each; the absence of notes in front of the British trio, compared to sheaves of papers on the EU side, sparked scathing jibes from critics in Britain who see Prime Minister Theresa's May's government as divided and badly unprepared.

Other meetings involved up to 10 officials on each side, getting "down to work" as Davis said, and well aware that a pressing March 2019 deadline may crimp their summer holidays.

"They have coffee," one source said. "And cookies, too."

The first objective is to show enough progress on a divorce package so that the EU will agree to open talks on a future free trade deal.

The three priorities are commonly referred to as citizens' rights, financial settlement and borders with Ireland. There are also a host of other loose ends that both sides want to see tied up legally to avoid a chaotic limbo on Brexit Day in March 2019.

London and Brussels agree that some 4.5 million expatriates who live on either side of a new EU-UK border, two-thirds of them in Britain, should not lose rights they have as EU citizens.