* Key budget cuts talks, choice for president in coming days
* Dogged by range of problems, Merkel's authority at stake
By Madeline Chambers
BERLIN, June 3 (Reuters) - Germany's Angela Merkel has hit a low point in her chancellorship, and her decisions in the coming days on tackling the budget deficit and who to back as president may determine whether she bounces back or is politically doomed.
Beset by a slump in popularity and a drubbing in a state election last month, and accused at home and abroad of weak leadership in the euro zone debt crisis, Merkel has also seen two high profile allies quit unexpectedly within a week.
This weekend, however, she has the opportunity to turn things around.
On Sunday and Monday she meets cabinet ministers to thrash out how to cut the bulging budget deficit, and she is also about to decide on a candidate for president after incumbent Horst Koehler's shock resignation this week.
"Merkel is standing on the edge of an abyss. But she has a chance to step back from it if she takes the right decisions in the next few days," said Nils Diederich, a politics professor at Berlin's Free University. He put her chances of success at about 50-50.
A poll this week put support for Merkel's conservatives on 30 percent, their lowest in nearly four years, and support for her Free Democrat (FDP) coalition partners has roughly halved since September's election.
A plethora of problems has made her look vulnerable and given ammunition to rivals and even some in her own party who accuse her of weak leadership during times of crisis.
"Quite simply, the chancellor is in a crap position," a Merkel aide told the Sueddeutsche Zeitung daily, while Bavarian health minister Markus Soeder said she was facing "days of reckoning".
BUDGET CRUNCH
Most crucial will be how she tackles the state finances.
Although the budget deficit in Europe's biggest economy won't be as large as those in many euro zone states this year, the government expects it to top five percent of gross domestic product -- significantly above the EU cap of three percent.
Merkel and her hard-nosed Finance Minister Wolfgang Schaeuble have already disappointed many voters -- and their FDP partners -- by scrapping planned tax cuts for 2011 and 2012.
They have warned of tough times ahead and say savings and spending cuts are the order of the day. They are bound by law to cut the structural deficit by 10 billion euros and the June 6-7 talks will agree the basis of the government's 2011 budget.
"The euro zone crisis has made voters worry about jobs and inflation and they are irritated Merkel has given little sense of direction. This makes the weekend meeting on budget cuts pretty crucial," Manfred Guellner, who heads polling group Forsa, told Reuters.
For economists and voters alike, the key question will be whether the government backs spending cuts or tax increases -- or opts for a combination of the two.
Possible increases in value-added tax and tobacco duty could fuel criticism that Merkel is endangering a fragile economic recovery and experts say she must persuade her ministers to slash spending.
"Merkel needs an austerity drive. If ministers agree to budget discipline without tax hikes, she could present it as a breakthrough and her position could stabilise," said Diederich.
He said it was unclear whether she would win the necessary backing of ministers from the FDP and Bavarian Christian Social Union (CSU) who did not want to look to be bowing to pressure.
Some analysts say that, to regain the initiative, Merkel needs a flagship policy comparable with former Social Democrat Chancellor Gerhard Schroeder's "Agenda 2010" labour market reform which defined his second term.
But Merkel is aware of the risks inherent in an austerity package. The deeply unpopular Agenda 2010 helped the economy by cutting welfare costs but also heralded Schroeder's downfall.
Merkel biographer Gerd Langguth said the cautious chancellor may try to play down her package to avoid scaring off voters. "Merkel is pragmatic and unideological. I doubt she will use an ideological headline to sell her policy," he said.
MOUNTAIN OF PROBLEMS
Another urgent question for Merkel is who she can get to succeed Koehler as president after his shock resignation.
Critics view his departure as a defeat for Merkel. They say she failed to back Koehler when he came under fire over comments he made about the army, even though he was her own candidate for president in 2004.
Merkel said she tried to persuade Koehler to stay and Spiegel Online reported that she warned him that resigning could trigger a crisis, and even shake public faith in state institutions.
His resignation, coupled with last week's exit from politics of Hesse state premier Roland Koch, a prominent ally who leaves a gap on the right of the conservative camp, has given the impression of rats abandoning a sinking ship, say commentators.
Merkel has been left scrambling to find a successor who will be elected by a special parliamentary assembly on June 30.
She may well get her choice -- whoever it ends up being -- through the assembly, where her coalition has a majority, but any hiccup would undermine her further.
Compounding her troubles is the need to persuade opposition parties to back her policies in the Bundesrat upper house as her centre-right coalition lost its majority there after a vote in the state of North Rhine-Westphalia last month.
The 2011 budget has to get through the Bundesrat, as do plans to extend the lives of some nuclear power stations.
Commentators say now is Merkel's chance to prove she is more than a fair weather chancellor. In her trademark matter-of-fact style, she has dismissed suggestions she isn't up to the task.
"You grow according to the challenges you are confronted with," Merkel told a television interviewer this week.
She has little option but to do just that. (Additional reporting by Andreas Rinke; editing by David Stamp)