* Confusion over prime minister adds to Egypt's crisis
* Liberal ElBaradei nominated until Islamists objected
* Political progress key after wave of deadly protests
Egypt's political transition after President Mohamed Mursi was ousted by the military stumbled at the first hurdle, after the choice of liberal politician Mohamed ElBaradei as interim prime minister was thrown into doubt by Islamist objections.
ElBaradei's nomination had been confirmed by several sources and state media on Saturday, but just before midnight a presidential spokesman told reporters that the prime minister had not in fact been chosen.
The abrupt U-turn came amid opposition to the appointment by the Nour Party, Egypt's second Islamist force after Mursi's Muslim Brotherhood movement, highlighting the challenge the military faces in finding consensus among liberals and conservatives on who should run the country.
Clashes between tens of thousands of pro- and anti-Mursi protesters swept the Arab world's most populous nation on Friday and at least 35 people were killed and more than 1,000 wounded.
The violence, which saw rival factions fighting street battles in central Cairo and many others cities and towns, underlined the pressing need for a swift political solution seen as inclusive to all.
"We extend our hand to everyone, everyone is a part of this nation," the spokesman told reporters. "The Muslim Brotherhood has plenty of opportunities to run for all elections including the coming presidential elections or the ones to follow."
Minutes after he spoke, state media reported that the public prosecutor ordered that four top Brotherhood leaders held this week be detained for a further 15 days on accusations that they incited violence against protesters.
The four included Saad el-Katatni, head of the group's political wing, and Khairat El-Shater, its political strategist.
ISLAMIST SUPPORT CRUCIAL
The Brotherhood has said it wants nothing to do with the military's plans for a new interim government. It believes Mursi should be reinstated, and has pledged to keep protesting until he is.
The Nour Party, however, had agreed to the army-backed transition plan leading to new elections. Its withdrawal from the process would strip that plan of vital Islamist support.
And following the Nour rejection, the interim administration headed by Adli Mansour delayed naming the new prime minister.
On Sunday, people were still reeling from one of the bloodiest days in over two years of tumultuous upheaval since autocrat Hosni Mubarak, who ruled for 30 years, was toppled in a popular uprising that was part of the 2011 "Arab Spring".
The Brotherhood called for another day of protest on Sunday, meaning that relative calm on Saturday may prove to be only a temporary lull.
Huge protests were staged on June 30 to pressure Mursi into resigning amid growing anger at economic stagnation and the perception among many that the Brotherhood was seeking to take control of every part of the state - a charge it fiercely denies.
Millions took to the streets to cheer his ouster on Wednesday, but for many Islamists it was a bitter reversal that raised fears of a return to the suppression they endured for decades under autocratic rulers.
The military said it had not carried out a coup, but merely enforced the will of the people.
The events of the last week have raised alarm among Egypt's allies in the West, including main aid donors the United States and the European Union, and in Israel, with which Egypt has had a U.S.-backed peace treaty since 1979.
On Saturday, President Barack Obama condemned the violence, and said the United States was not working with any particular party or group in Egypt.
Washington has not condemned the military takeover or called it a coup, prompting suspicion within the Brotherhood that it tacitly supports the overthrow of the country's first freely elected president.
Obama has ordered a review to determine whether annual U.S. assistance of $1.5 billion, most which goes to the Egyptian military, should be cut off as required by law if a country's military ousts a democratically elected leader.
Egypt can ill afford to lose foreign aid. The country appears headed for a looming funding crunch unless it can quickly access money from overseas. The local currency has lost 11 percent of its value since late last year.
Newspapers on Saturday quoted ElBaradei as saying he expected Gulf Arab monarchies that were hostile to the Brotherhood's rule to offer financial support to the new authorities.
Only gas-rich Qatar provided substantial funds to Mursi's government, totalling $7 billion in loans and grants. Turkey and Libya also provided smaller loans and deposits.