* After backing govt, US signals mediation bid with Arab states
* Government must let protesters demonstrate in peace-Kerry
* Brotherhood escalates street campaign, calls more protests
* ElBaradei says he opposing those bent on crushing Brotherhood
* Rights group alleges torture by Mursi supporters (Adds details of clash, Amnesty report)
By Tom Perry and Michael Georgy
CAIRO, Aug 2 (Reuters) - The United States said on Friday it would work with other nations to resolve Egypt's crisis peacefully, injecting new energy into a push to end a bloody standoff since the overthrow of Islamist President Mohamed Mursi.
A day after saying the army had restored democracy by removing Mursi, U.S. Secretary of State John Kerry urged Egyptian authorities to give demonstrators the space to protest in peace - a warning against dispersing pro-Mursi sit-ins.
"We will work very, very hard together with others, in order to bring parties together to find a peaceful resolution that grows the democracy and respects the rights of everybody," Kerry said before a meeting United Arab Emirates Foreign Minister Sheikh Abdullah bin Zayed in London.
The meeting appeared to signal a new diplomatic effort to end the crisis in which more than 300 people have been killed. The army removed Mursi and his Muslim Brotherhood from power on July 3 in response to mass protests against his rule.
With the European Union already mediating, the new push will rely on the United Arab Emirates to work with the army-backed interim government and Qatar, which supported the Mursi administration, to liaise with the Brotherhood.
Analysts say civilian members of the interim cabinet are trying to promote a political solution despite resistance from security services that want to crack down on the Brotherhood, encouraged by an outpouring of public anger at the movement.
Mohamed ElBaradei, vice president in the new administration, said he was lobbying for talks with the Brotherhood, while others advocated crushing it.
"People are very angry with me because I am saying, 'Let's take time, let's talk to them'. The mood right now is, 'Let's crush them, let's not talk to them'," he said. "I hope the Brotherhood understands that time is not on their side. I'm holding the fort, but I can't hold it for very long."
Egypt is more polarised than at any time since the downfall of autocratic president Hosni Mubarak in 2011, complicating mediation efforts in a pivotal Arab state fraught with unrest.
The Brotherhood, decrying what it sees as a coup against the country's first freely-elected head of state, escalated its protest campaign by announcing two new sit-ins and three marches to sensitive security facilities.
Its supporters clashed with police during a protest near a complex of television studios outside Cairo. Police fired tear gas to disperse the crowd. State media reported army helicopters overhead and said two policemen were wounded by birdshot in what it described as an attempt to storm the complex.
The Brotherhood said the security forces had fired tear gas on peaceful protesters. Seven Mursi supporters were also injured, security sources said.
State TV reported that the Interior Ministry would impose a cordon around two large pro-Mursi Cairo sit-ins within 48 hours and that the police did not want to break them up by force.
Following the deaths of around 80 Mursi supporters on Saturday when security forces opened fire near one of the sit-ins, government pledges of action have raised concerns of more casualties. By pushing back any move to break up the sit-ins, the government has given the mediation effort a chance.
With the United States supplying Egypt with $1.3 billion in military aid each year and the United Arab Emirates having pledged $3 billion to the new administration, the countries may be able to help force a compromise with Qatar's help.
In London, Sheikh Abdullah said a peaceful resolution required "inclusive dialogue".
The government has drawn up a transition plan envisaging parliamentary elections that will start in about six months.
But the Brotherhood protests are threatening to rob the government of a semblance of normality it needs to revive an economy which is in deep in crisis.
SISI SEES NEED FOR POLITICAL SOLUTION
Mursi has been in detention since he was deposed and is facing a judicial inquiry into accusations of murder and conspiring with the Palestinian group Hamas.
The authorities have also rounded up many other Brotherhood leaders accused of inciting violence, feeding international fears of a plan to uproot a group that was suppressed for decades until Mubarak's overthrow. The government accuses Mursi's supporters of taking up arms, alleging they engage in terrorism.
ElBaradei, former head of the United Nations nuclear watchdog, outlined ideas for a political deal that might include a pardon for Mursi and guarantees that the Brotherhood would have a place in political life. He said army chief Abdel Fattah al-Sisi understood the need for a political solution.
"But of course he has a responsibility to protect the country in terms of security. And the army is on the edge."
He said dialogue was the way to end the Brotherhood sit-ins. The government has promised to deal with protests it sees as a threat to national security. "I do not want to see any more bloodshed. Nobody wants that," ElBaradei said.
"They need to cooperate," he added, in reference to the Brotherhood. "But they need of course to feel secure, they need immunity, they need to feel that they are not excluded. It's things we are willing to provide."
He added that Sisi, who has gained enormous popularity since deposing Mursi, was not thinking of running for president.
"WE'RE NOT TERRORISTS"
The biggest sit-in is in northeast Cairo, where several thousand Mursi supporters have been camped out for more than a month in a protest that at times swells to tens of thousands.
"We are here with our wives and children. We don't want violence," said Ali el-Shishtawi, a government employee at the sit-in. "We're not afraid. We're not terrorists like they say."
Amnesty International issued a report saying Mursi supporters had tortured some of their political rivals, saying anti-Brotherhood protesters had reported being been captured, stabbed, beaten and subject to electric shocks.
It said eight bodies had arrived at the morgue in Cairo bearing signs of torture, five of which had been found near pro-Mursi sit-ins, and called for an investigation.
The new government gained a U.S. seal of approval late on Thursday when Kerry said the army had been "restoring democracy" when it toppled Mursi - Washington's strongest endorsement yet for the new leadership.
"The military was asked to intervene by millions and millions of people, all of whom were afraid of a descent into chaos, into violence," Kerry told GEO TV in Pakistan. "And the military did not take over, to the best of our judgment - so far."
Washington's efforts to avoid calling Mursi's overthrow a "military coup" has left it open to charges of sending mixed messages about events in Egypt, long a bulwark of U.S. Middle East policy.
Mohamed Ali Bishr, a senior Brotherhood leader, said the movement was disappointed by Kerry's statement. "The United States is a country that speaks of democracy and human rights and they say something like that. I hope that they rethink their position and correct it," he told Reuters. (Additional reporting by Maggie Fick and Noah Browning in Cairo and Lesley Wroughton in London; editing by David Stamp)