By Asma Alsharif and Tom Perry
CAIRO, July 1 (Reuters) - An outpouring of popular anger that put millions of Egyptians on the streets subsided on Monday but the country's leaders seem no closer to ending a political stalemate that has left some looking to the army to break the deadlock.
Literally under fire overnight, as its national headquarters were set alight, President Mohamed Mursi's Muslim Brotherhood remains besieged by liberals and others who want Mursi to resign. Even some fellow Islamists say he must make concessions.
Cairo's Tahrir Square, which saw crowds on Sunday greater than any since the revolution of 2011, was quiet as the working day began. Only a few hundred were camping out. The same was true across the country. But at the Brotherhood compound in Cairo, guards continued to shoot at youths in the streets around.
Late on Sunday, Mursi's movement said the building, on a hill overlooking the capital, was surrounded by dozens of men firing shotguns and throwing rocks and petrol bombs. Reuters journalists saw flames in parts of the building and exchanges of fire between those inside and outside. On Monday morning, a Reuters journalist saw two young men hit by gunfire outside.
Brotherhood leaders complained that police failed to appear to protect their headquarters - a mark of the movement's difficulties in controlling the security services since Mursi became Egypt's first freely elected leader a year ago on Sunday.
No police were present on Monday morning. Staff at a nearby hospital said two people had been killed and 45 treated for gunshot wounds from the incident overnight. On Monday morning, at least a further six people had been wounded, medics said.
The demonstrations, which brought half a million people to Tahrir Square and a similar crowd in the second city, Alexandria, were easily the largest since the Arab Spring uprising that ousted Hosni Mubarak two and half years ago.
"LIARS AND LOSERS"
Mursi, stayed out of sight throughout the protests but acknowledged through a spokesman that he had made mistakes, adding that he was working to fix them and was open to dialogue.
He showed no sign of quitting.
Senior Brotherhood official Essam El-Erian sounded upbeat on Facebook: "There was no civil war, as the liars advertised ... and there will be no military coup as the losers want.
"There is no alternative to unconditional dialogue to reach an understanding about forthcoming parliamentary elections."
An aide to Mursi outlined three ways forward: parliamentary elections, which he called "the most obvious"; national dialogue, which he said opponents had repeatedly rejected; and third, early presidential elections, as demanded by protesters.
But that, he said, "simply destroys our democracy".
The massive protests showed that the ruling Muslim Brotherhood has not only alienated liberals and secularists by seeking to entrench Islamic rule but has also angered millions of ordinary Egyptians with economic mismanagement.
Tourism and investment have dried up, inflation is rampant and fuel supplies are running short, with power cuts lengthening in the summer heat.
Protest organisers called on Egyptians to keep occupying central squares across the country in a campaign of peaceful civil disobedience until Mursi quits. They plan a march on Tuesday evening to the place where Mursi has been working if he has not stepped down by 5 p.m. (1500 GMT) that day.
If protesters maintain their pressure, possibly returning to main squares in the evenings, the spotlight will be on the army.
It displayed its neutrality on Sunday, making goodwill gestures to the protesters after urging feuding politicians last week to cooperate to solve the nation's problems.
Some uniformed policemen marched among protesters in Cairo and Alexandria, chanting "the police and the people are one", and several senior officers addressed the Tahrir Square crowd.
That cast doubt on whether Mursi could rely on the security forces to clear the streets if he gave the order.
Diplomats said the army, which ruled uneasily during the transition from Mubarak's fall to Mursi's election, had signalled it was deeply reluctant to step in again, unless violence got out of hand and national security was at stake.
While the main demonstrations were mostly peaceful and festive in atmosphere, seven people were shot dead in clashes in the central cities of Assiut, Beni Suef and Fayoum and outside the Brotherhood's Cairo headquarters. The Health Ministry said 613 people were wounded in street fighting around the country.
Women's activists said at least 43 women, including a foreign journalist, suffered organised sexual assaults by gangs of men during the Tahrir Square rally.
The opposition National Salvation Front coalition of liberal, secular and left-wing parties declared victory, saying the masses had "confirmed the downfall of the regime of Mohamed Mursi and the Muslim Brotherhood".
Opposition leaders, who have seen previous protest waves fizzle after a few days in December and January, were to meet on Monday afternoon to plot their next move.
The United States and the European Union have urged Mursi to share power with the opposition, saying only a national consensus can help Egypt overcome a severe economic crisis and build democratic institutions. (Writing by Paul Taylor and Alastair Macdonald; Editing by Peter Graff)