* Negotiators meet a year after Brexit referendum shock
* Davis wants "deep and special partnership" with EU
* May's election debacle leaves British plans unclear
* EU heartened by new president Macron's sweep in France (Adds details, lunch menu, gifts)
BRUSSELS, June 19 (Reuters) - Britain's negotiators came to Brussels seeking a "new, deep and special partnership with the European Union" on Monday as talks on the unprecedented British withdrawal from the bloc finally got under way.
A beaming Brexit Secretary David Davis, a veteran campaigner against EU membership, told a somber Michel Barnier, the EU's chief negotiator, that his team aimed to maintain a "positive and constructive tone" during "challenging" talks ahead in the hope of reaching a deal that was in the interests of both sides.
A year after Britons shocked the continent by voting on June 23 to cut loose from their main export market, new debate within Prime Minister Theresa May's cabinet on precisely what kind of trading relationship to pursue has perplexed EU leaders, who warn time is tight to agree terms before Britain leaves in 2019.
"We must first tackle the uncertainties caused by Brexit," said Barnier, a former French minister, as he greeted Davis at the European Commission's Berlaymont Building headquarters.
Those were, he said, the rights of expatriate citizens and problems of a new EU-UK border, notably cutting across Ireland. He did not mention a third EU priority - that Britain settle a bill of tens of billions of euros before it leaves in 21 months.
That financial issue is already a bone of contention, as is Brussels' refusal to discuss a new free trade deal until after it is resolved. May, whose future is uncertain after she lost her Conservative majority in an election this month, has insisted that trade talks start immediately and run in parallel.
While Barnier insists on the "sequencing" of talks, so that trade negotiations cannot start until probably January, finding a way to avoid a "hard" customs border for troubled Northern Ireland may well involve some earlier discussion of the matter.
A bigger problem may be for British negotiators to resolve what trade relationship they want. While "Brexiteers" like Davis have strongly backed May's proposed clean break with the single market and customs union, finance minister Philip Hammond and others have this month echoed calls by businesses for less of a "hard Brexit" and retaining closer customs ties.
The bloc has expanded steadily since first formed as the European Economic Community in 1957 by France, West Germany, Italy, the Netherlands, Belgium and Luxemburg. It currently numbers 28 members. Never before has a country sought to leave.
Davis, noting shared security threats for governments across Europe hours after a van rammed worshippers at a London mosque, said: "There is more that unites us than divides us.
"We are ... determined to build a strong and special partnership between ourselves, our European allies and friends."
Officials on both sides played down expectations for what can be achieved in one day. EU diplomats hope this first meeting, and a Brussels summit on Thursday and Friday where May will encounter - but not negotiate with - fellow EU leaders, can improve the atmosphere after some spiky exchanges.
Before lunching on Belgian asparagus, red mullet and meringue cake, the two exchanged gifts that homed in on a shared love of hiking: Davis received a walking stick from Barnier's native Savoy Alps and Barnier a first edition of a French mountaineer's Himalayan memoir - "Regards vers Annapurna."
Davis's agreement to Monday's agenda led some EU officials to believe May's government may at last be coming around to Brussels' view of how negotiations should be run.
After Davis and Barnier met over lunch in the Commission's top floor dining rooms, their teams broke up into "working groups" that will be charged with handling specific areas of talks that the EU expects to take place for a week every month.
Barnier said he was hoping to have a clearer timetable by the end of the day. He has said a divorce deal should be ready by October next year to give time for parliamentary approval. With or without a deal, Britain will be out of the EU on March 30, 2019. EU leaders want May to lay off threats that she would walk out and leave a chaotic legal limbo for all Europeans.
But EU leaders, including German Chancellor Angela Merkel and French President Emmanuel Macron, are also determined not to make concessions to Britain that might encourage others to quit.
When 52 percent of British voters opted for Brexit, some feared for the survival of a Union battered by the euro crisis and divided in its response to chaotic immigration. The election of the fervently europhile Macron, and his party's sweep of the French parliament on Sunday, has revived optimism in Brussels. (Editing by Janet Lawrence)