CAIRO (Reuters) - Cairo police fought protesters demanding an end to army rule for a third day on Monday and the death toll rose to 33, with many victims shot, in the worst violence since the uprising that toppled President Hosni Mubarak.

As midnight approached, about 20,000 people packed Tahrir Square, the epicentre of the anti-Mubarak revolt early this year, and thousands more milled around in surrounding streets.

"The people want the fall of the marshal," they chanted, referring to Field Marshal Mohamed Hussein Tantawi, Mubarak's defense minister for two decades and head of the army council.

In a late-night statement, the ruling council urged calm and called for crisis talks with political forces to find a way forward. The council voiced its "deep regret for the victims in these painful incidents", state news agency MENA said.

"It called on all sections of the nation to show the greatest degree of self-restraint so that the matter does not lead to more victims and casualties," the agency added.

The military council did not say whether it would accept the resignation of the cabinet, tendered on Sunday. A military source said it was seeking agreement on a new prime minister.

The resignation of the cabinet, in office since March, was another blow to the military council's authority and casts further uncertainty on Egypt's first free parliamentary elections in decades, which are due to start next Monday.

Clashes flared in side-streets near Tahrir. Witnesses said looters, not necessarily connected to the protests, had attacked the American University in Cairo and other buildings.

Security forces also battled about 4,000 demonstrators in the port city of Ismailiya on the Suez Canal, witnesses said. Tear gas was also fired at about 2,000 protesters in the northern coastal city of Alexandria.

BULLET WOUNDS

Protesters have brandished bullet casings in Tahrir Square, where police used batons and tear gas to try to disperse demonstrators on Saturday. Police deny using live fire.

Medical sources at Cairo's main morgue said 33 corpses had been received since Saturday, most with bullet wounds. The Health Ministry put the toll at 24 dead and 1,250 wounded.

"I've seen the police beat women my mother's age. I want military rule to end," said protester Mohamed Gamal, 21.

Army generals were feted for their part in easing Mubarak out, but hostility to their rule has hardened since, especially over attempts to set new constitutional principles that would keep the military permanently beyond civilian control.

Police attacked a makeshift hospital in Tahrir Square after dawn but were driven back by protesters hurling chunks of concrete from smashed pavements, witnesses said.

"Don't go out there, you'll end up martyrs like the others," protesters told people emerging from a metro station at Tahrir Square.

The violence casts a pall over the first round of Egypt's staggered and complex election process, which starts on November 28 in Cairo and elsewhere. The army says the polls will go ahead.

INTERNATIONAL CONCERN

The United States called for restraint on all sides and urged Egypt to proceed with elections despite the violence.

"The United States continues to believe that these tragic events should not stand in the way of elections," White House spokesman Jay Carney said.

European Union foreign policy chief Catherine Ashton echoed that message and said the EU was keen to monitor the polls.

"The Egyptian authorities have been very clear that they wish to conduct these elections themselves. We believe it would give credibility to them to have international observation," she told British lawmakers in London.

U.N. Secretary-General Ban Ki-moon deplored the loss of life and called on the transitional authorities "to guarantee the protection of human rights and civil liberties for all Egyptians, including the right to peaceful protest".

In an apparent sop to protesters, the army council issued a law to bar from political life "those who work to corrupt political life and damage the interests of the nation".

The announcement was unlikely to satisfy political parties and activists who have called for a blanket ban on former members of Mubarak's now defunct National Democratic Party.

"The council is out of step with the people," said activist Mohamed Fahmy, describing the new law as a "meaningless move".

Some Egyptians, including Islamists who expect to do well in the vote, say the ruling army council may be stirring insecurity to prolong its rule, a charge the military denies.

Political uncertainty has gripped Egypt since Mubarak's fall, while sectarian clashes, labor unrest, gas pipeline sabotage and a gaping absence of tourists have paralysed the economy and prompted a widespread yearning for stability.

ECONOMIC WOES

The instability could accelerate Egypt's slide toward a currency crisis, forcing a sharp depreciation of the Egyptian pound in the next few months and conceivably prompting Cairo to impose capital controls, analysts said.

"The violence and political noise is going to erode whatever confidence was left in the Egyptian economy, and may result ... in an acceleration of capital outflows," said Farouk Soussa, Middle East chief economist at Citigroup.

The military plans to keep its presidential powers until a new constitution is drawn up and a president is elected in late 2012 or early 2013. Protesters want a much swifter transition.

The army said on Monday it had intervened in central Cairo to protect the Interior Ministry, not to clear demonstrators from nearby Tahrir Square, whom it also offered to protect.

The Interior Ministry, in charge of a police force widely hated for its heavy-handed tactics in the anti-Mubarak revolt, has been a target for protesters demanding police reform.

The latest street clashes show the depth of frustration, at least in Cairo and some other cities, at the pace of change.

Liberal groups are dismayed by the military trials of thousands of civilians and the army's failure to scrap a hated emergency law. Islamists eyeing a strong showing in the next parliament suspect the army wants to curtail their influence.

Analysts say Islamists could win 40 percent of assembly seats, with a big portion going to the Muslim Brotherhood.

The Revolution Youth Coalition, an activist group, called for a "million-man march" in Tahrir on Tuesday to back demands for a new national salvation government to run the country in the transition phase, instead of the military council.