* Army chief wants Bouteflika out on health grounds
* Protesters reject managed exit, army involvement
* Fears of further turmoil in gas, oil producer (Adds details, colour)
By Lamine Chikhi and Hamid Ould Ahmed
ALGIERS, March 27 (Reuters) - Algeria's biggest union and an influential party on Wednesday backed an army call for President Abdelaziz Bouteflika to quit in a managed exit plan that was quickly rejected by protesters demanding the overthrow of the entire political elite.
The statement from the National Rally for Democracy (RND), a member of the ruling coalition, came a day after the military - Algeria's traditional kingmakers - said Bouteflika should be declared unfit for office.
The General Union of Algerian Workers (UGTA), long a staunch supporter of the president, also said it supported the army call for Bouteflika to step down.
The announcements by three pillars of the establishment were a clear signal that the 82-year-old president - who has rarely appeared in public since suffering a stroke in 2013 - has little to no chance of staying in power in the North African country, an oil and gas producer.
The army has been patiently waiting for the right moment to intervene, after winning over Bouteflika's allies in a process that has emptied out his inner circle, in the hopes of a smooth transition period, political sources said.
But the leaders of five weeks of mass protests fuelled by anger over alleged corruption, nepotism and economic mismanagement said the plan still did not go far enough, risking a standoff with the military.
"Protests will continue... Algerians' demands include a change of the political system," Mustapha Bouchachi, a lawyer and activist, told Reuters.
"The implementation of Article 102 (the part of the constitution that covers declaring a president unfit for office) means that the symbols of the system will oversee the transition period and organise presidential elections," he said.
Even if both sides dig in, no Algerians want to risk returning to the dark days of the 1990s, when the army's cancellation of elections that Islamists were poised to win triggered a civil war that killed 200,000 people.
Back then, Islamists had marched in the streets chanting "no laws, no constitution but Allah" under the watchful eye of the military. No women had taken part and the men were bearded, in sharp contrast to the protests which erupted five weeks ago.
These demonstrations have been peaceful, with people singing, laughing and taking selfies as they wave banners calling for peaceful political change. Some even bring lunch to the demonstrations. The military has stayed in its barracks.
The army chief has warned he will not tolerate chaos, but he also praised Algerians for pursuing "noble aims".
Protesters have repeatedly said they would reject any orchestrated succession in politics and want a transition which will lead to a government by consensus, but there are no signs they will resort to violence to reach their goals.
"We want a real democracy, not a facade of a democracy," said postal worker Zakaria Jaziri 26.
State bank employee Djamel Hadidi, 37, said: "We welcome the army's initiative but we do not want Bouteflika's men to govern us until the next election."
The army's powerful chief of staff, Lieutenant General Ahmed Gaed Salah, told officers in a speech broadcast on Tuesday that the solution to the crisis would be the departure of the president on health grounds.
Salah called on Algeria's constitutional council to rule whether Bouteflika was fit for office. Such a ruling would have to be ratified by a two-thirds majority of members in the two houses of parliament.
Under Article 102 of the constitution, the chairman of parliament's upper house, Abdelkader Bensalah, would serve as caretaker president for at least 45 days after Bouteflika's departure.
The leader of the RND party, Ahmed Ouyahia, urged Bouteflika to resign under Article 102, which also covers presidential resignations.
Even if Bouteflika quits, there is no obvious replacement who could be acceptable to all sides.
For years, rumours have swirled about potential successors, but no single credible candidate has emerged with the backing of the military and the political and security establishment who is not at least 70.
Facing the biggest challenge to his rule, Bouteflika reversed a plan to seek a fifth term, postponed elections and promised to introduce greater freedoms.
But he stopped short of stepping down, infuriating Algerians who want to do away with veterans of the 1954-1962 war of independence against France, military officers and business tycoons who have dominated for decades.
"Is there a risk of radicalisation or confrontation if demonstrators reject the army approach? This is a hypothesis that cannot be totally excluded," said Louisa Dris, professor of political science at Algiers University.
The stakes are high - Algeria is a leading member of OPEC and a top gas supplier to Europe, though so far oil and gas output appears unaffected by the unrest, an International Energy Agency (IEA) official said on Tuesday.
Algeria is also regarded by Western states as a partner in counter-terrorism, a significant military force in North Africa and a leading diplomatic player in efforts to resolve crises in neighbouring Mali and Libya. (Additional reporting by Abdelaziz Boumzar; Writing by Michael Georgy, Editing by Andrew Heavens, William Maclean, Gareth Jones)
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