* Naming interim cabinet to take several more days
* Muslim Brotherhood calls for more protests
* Military orders arrest of Brotherhood leaders
* Egypt deeply divided over killings, deadly protests
Political infighting threatened to stall Egypt's transition plans on Thursday, as the military cracked down on Muslim Brotherhood leaders it blames for inciting a clash in Cairo in which troops shot and killed 53 protesters.
Monday's violence between supporters of ousted President Mohamed Mursi, Egypt's first freely elected leader toppled by the army last week, and soldiers at a military compound has opened deep fissures in the Arab world's most populous country.
On Wednesday, Egypt's public prosecutor ordered the arrest of Muslim Brotherhood leader Mohamed Badie and several other senior Islamists, evoking memories of when the movement was repressed under autocratic former Egyptian leader Hosni Mubarak, who was ousted in 2011.
Mursi's Muslim Brotherhood called for protest marches on Friday, when noon prayers are held in mosques, raising the risk of more violence after fighting between rival factions swept Egypt last week and killed 35 people.
Muslim Brotherhood spokesman Gehad El-Haddad said the leaders had not been arrested and some were still attending a protest vigil at Rabaa Adawiya mosque, where thousands of supporters have camped out for the past two weeks despite punishing heat.
He said the charges against them of inciting violence were "nothing more than an attempt by the police state to dismantle the Rabaa protest."
"What can we do?" he asked. "In a police state, when the police force are criminals, the judiciary are traitors and the investigators are the fabricators, what can one do?"
Egypt's 84 million people are increasingly divided between those who rallied on June 30 to demand Mursi's resignation and angry Islamists who say their democratic rights have been crushed in what they call a military coup.
Separately, the youth-led Tamarud group, which coordinated the mass protests against Mursi centred in Cairo's Tahrir Square, invited its followers to rally there, also on Friday, in a festive celebration of the Muslim fasting month of Ramadan.
Ramadan, normally a time of celebration, falls this year under the dark shadow of a crisis that has left Egyptian society more divided than at any time in its modern history.
The situation in Cairo and other cities has calmed significantly since last week's protests and Monday's clash in which four security personnel were also killed.
Hazem el-Beblawi, the interim prime minister, told Reuters he expects the transitional cabinet to be in place early next week as he seeks to implement the military-backed "road map" that envisages new parliamentary elections in about six months.
Beblawi acknowledged that it will be a challenge to find a cabinet line-up with universal support. "I don't believe that anything can have unanimous approval," he said.
The economist has indicated he would be open to offering cabinet posts to Islamists, including Muslim Brotherhood figures. The Muslim Brotherhood says it will have nothing to do with a government of what it calls a fascist coup.
WORDING OF A NEW CONSTITUTION
Another potential stumbling block in a political process vital to restoring stability is wrangling among Islamist and liberal parties over the wording of a new constitution, in particular articles defining the role of Islamic sharia law.
The liberal National Salvation Front may be willing to compromise in order to prevent further delays to the process, a senior party leader said.
The unrest after Mursi's ouster on July 3, initially greeted with jubilation by hundreds of thousands of people across the sprawling capital, has alarmed Western donors and Israel, which has a 1979 peace treaty with Egypt.
The United States, treading a careful line, has neither welcomed Mursi's removal nor denounced it as a coup. Under U.S. law, a coup would require the United States to halt aid, including the $1.3 billion it gives the army each year.
The Brotherhood's downfall has been warmly welcomed by three of the rich Arab monarchies of the Gulf, who showered Cairo with aid to prop up the collapsing economy.
Kuwait promised Egypt $4 billion in cash, loans and fuel on Wednesday, a day after Saudi Arabia pledged $5 billion and the United Arab Emirates offered $3 billion.
In the early hours of Thursday, several hundred Islamist protesters marched to the presidential palace. Demonstrators and police remained calm and the two sides engaged in conversation.
"It is our right to say we don't acknowledge the military coup," said one of the protesters, addressing soldiers outside the palace in a good-natured exchange.
Both sides in Egypt have become more anti-American in recent weeks. Mursi's opponents say President Barack Obama's administration supported the Muslim Brotherhood in power, while Mursi's supporters believe Washington was behind the plot to unseat him.
"Obama supports democracy, but only if it goes to those who aren't Islamists," the heavily bearded El-Sayyed Abdel Rabennabi said at the Rabaa vigil.
On Tahrir Square, the animosity appears just as fierce.
"America made an alliance with the Brotherhood against the Egyptian people," said Tawfiq Munir, waving a placard reading "We are the coup" at one recent rally there.
"Now the Brotherhood are fighting us in the streets, fighting to take back power, and America is sitting on the fence," said the aircraft mechanic.