(Repeats story filed on Thursday, updating with EU agreement)
* EU finally moves to open talks on trade
* After unity, states eye varied commercial interests
* Britain courts continental capitals, separately
* "Divide and rule" tactics won't work, EU insists
BRUSSELS, March 23 (Reuters) - Signs of cracking in the European Union's common front on Brexit may encourage hopes in London that Prime Minister Theresa May can exploit continental divisions to secure British advantage.
But EU leaders and diplomats, as well as some British Brexit experts, argue that May risks disappointment in that quest, as Germany leads a push to prevent London unravelling the EU single market rules and the 27, mildly intoxicated by their success in the first round of talks, see unity benefiting all of them.
Rare harmony among member states has strongarmed Britain into largely accepting the EU's terms for its departure. But the negotiating stance for future trade, launched at a summit on Friday, has been preceded by a revival of nationalistic jostling among the 27 member states that is more typical of Brussels.
"So far, it's under control," one senior EU diplomat said of signs of disunity. "But it'll come for sure."
British officials make no secret of hoping to set allies against one another as London seeks maximum trading markets.
"The most vital thing, first of all, is to maintain the unity of the 27," Dutch Prime Minister Mark Rutte told a French newspaper before a meeting with President Emmanuel Macron this week that brought together a traditional free-trading ally of London with Britain's Gallic long-time rival for influence.
Despite their differences, EU governments surprised many in London -- and themselves -- by holding to a common line to force May into first agreeing to pay a hefty bill and grant sweeping rights for immigrants before she even gets to talk about trade.
The guidelines leaders endorsed on Friday to offer an "ambitious and wide-ranging free trade agreement" have generated little debate. "It's much easier to agree general political declarations," the senior EU envoy said. "But it'll get way more difficult when we sit down to detailing the exact solutions."
"DIVIDE AND RULE"
This week several countries secured a special agreement among the 27 to protect various airline and airport interests -- Spain's British-owned flag-carrier Iberia or Finland's layover hub for London-Asia flights -- and another pushed by Luxembourg to promise good treatment for London's banks and finance firms.
Another senior Brussels diplomat from a member state said that already smacked of "cherry-picking" -- the very sin the 27 accuse Britain of doing, to keep the best bits of EU membership.
North Sea fisheries are also brewing as a sore point.
May's negotiator, Brexit Secretary David Davis, has eschewed long meetings in Brussels in favour of touring the capitals of Europe, looking for allies in Britain's push for a "bespoke" deal that will keep much of its access to markets.
London is wary of being seen going behind the backs of the European Commission in Brussels and its negotiator Michel Barnier: "There is some sensitivity in the Commission about us trying to divide and rule," a British official said. "So we've made a point of getting around all of the states one by one."
London, the official said, was hoping for help in gaining EU financial services access from Luxembourg, which is wary of French efforts to freeze out the City of London in the hope of drawing bankers to Paris. Eastern European states, with less trade with Britain but more worries about Russia and EU subsidies, are also keen to keep London close as a military ally and potential continuing contributor to some Brussels budgets.
"GIVING, GIVING, GIVING"
But officials and diplomats from the east tend to bristle at suggestions they might be bought off to break ranks. The lesson of the past year, they say, is that unity works for everyone.
"Unity among the 27 has seen the British giving, giving, giving. We've got more than we ever dreamed of," said a senior aide to one of the eastern leaders attending the summit.
"Of course people will press for their own interests. It will get more difficult -- but unity seems to work."
That is the conclusion of a report last week by the Centre for European Reform in London: "A stance of complete unity worked in phase one of the talks," it said. "The Commission, Germany and France are in a strong position to persuade the others ... and the overwhelming interest among the 27 will remain ensuring the integrity of the single market."
Barnier and others warn Britain that sowing division will be counter-productive, raising a risk of running out of time to avoid chaos: "If we lose unity, we lose everything," a third senior EU diplomat said. "If we are not united, that won't help the UK -- things would just become blocked."
As an example of resolve, Brussels insiders point to how the 27 have rallied behind Ireland as it presses London to clarify border arrangements in the north -- a minor issue for most continentals compared to maintaining smooth trade with Britain.
Chancellor Angela Merkel rolled out a red carpet for Irish Prime Minister Leo Varadkar in Berlin this week in a timely display of solidarity. German exporters might fret about lost business, but Merkel aides insist, she has convinced most that Germany has much more to lose by letting Britain "cherry pick" trade deals which Merkel says could unravel the EU altogether. (Additional reporting by Elizabeth Piper in London and Gabriela Baczynska in Brussels Editing by Richard Balmforth)