Egyptians shout slogans Friday during an anti-President Mohammed Morsi protest in front of the presidential palace in Cairo. (Amr Nabil/AP)

Protesters armed with Molotov cocktails clashed with police outside the presidential palace in Egypt’s capital Friday evening, as demonstrations against President Mohamed Morsi carried into a second week in major cities across the country.

Local TV footage showed a chaotic scene, with trees along the palace walls consumed by flames. Later, riot police fired shotgun pellets, tear gas and powerful jets of water from fire hoses in an effort to push back protesters and extinguish the blazes. The presidency said in a statement that it would not hesitate to use force to protect state property.

At least one protester was shot dead, the Health Ministry confirmed, and local media reported dozens of injuries, mostly from tear-gas inhalation.

The crisis gripping Egypt has exposed a growing disconnect between the opposition movement’s political leadership and many of the young protesters in the streets. The latest violence came a day after top opposition figures held a dialogue session with Morsi’s backers in the Muslim Brotherhood and other conservative Islamists, later joining with them to issue a statement condemning violence.

Many of the protesters who began rallying outside the palace earlier in the day expressed skepticism about the dialogue effort, and some openly defied the call for nonviolence as they hurled firebombs at the complex from which Morsi administers the country.

TIMELINE: Key events in Egypt since the fall of Hosni Mubarak.

As fires raged Friday night, Morsi and the Muslim Brotherhood used Twitter to urge protesters to retreat and pointed fingers at the opposition. “We consider political forces that could have contributed, by instigating, to have full political responsibility, pending the results of an investigation,” Morsi tweeted as night fell.

Earlier Friday, smaller crowds in Cairo and key Suez Canal cities — where the recent violence has been most intense — had appeared to suggest that the crisis was abating, despite opposition calls for more marches. Protesters said blustery weather, fatigue and disagreements in their ranks over their movement’s goals stifled turnout after a week of clashes that left dozens dead and more than 1,000 injured.

For most Egyptians, the first day of the country’s weekend passed at its usual quiet pace.

The sheik of Al-Azhar, Egypt’s highest religious authority, hosted Thursday’s dialogue session, which was attended by leaders of the National Salvation Front (NSF), a loose alliance of liberal and secular opposition figures; the ultraconservative Salafists; and members of the Muslim Brotherhood. The joint statement yielded by the meeting denounced the use of violence by all parties.

But it was unclear what, if any, impact the dialogue had on Friday’s demonstrations. Many participants said they were disappointed in the NSF, which has called for a national unity government, and would keep protesting until Morsi was ousted.

The opposition movement has been plagued by divisions over goals and tactics, even as it widened in the past week on the back of a wave of popular anger over police brutality.

“There’s definitely a fundamental difference on whether the goal of protest and making demands is . . . to get concessions from Morsi or whether they want to force Morsi from power,” said Michele Dunne, director of the Rafik Hariri Center for the Middle East at the Atlantic Council, a Washington think tank.

Many protesters in Cairo on Friday described conflicting visions of a resolution to the crisis. Some called for a national unity government and peaceful protest. Others, citing bread-and-butter economic issues, chanted for Morsi’s dismissal. Members of the “black bloc,” a new, black-masked anarchist group that has advocated violence against the state, arrived at the palace protest wearing T-shirts that read “Revenge,” witnesses said.

Clashes also flared in the port city of Alexandria and the Nile Delta towns of Kafr el-Sheikh and Tanta. In the coastal city of Port Said, where more than 40 people have died in the past week’s clashes, some protesters called for an independent state.

“I’m not really interested in getting closer to this unity government,” said Hussein Abdel Ghani, a spokesman for the NSF who took to the streets Friday and said he had opposed Thursday’s dialogue. “I’m interested in pushing the regime to get back to the common ground of our revolution goals: freedom and social justice.”

The crisis, set off by protests Jan. 25 to mark the two-year anniversary of the start of the uprising that ousted Hosni Mubarak, has amplified pressure on Morsi and his backers in the Brotherhood amid an atmosphere of general insecurity and skyrocketing inflation.

Many Egyptians have accused Morsi, Egypt’s first democratically elected president, and the Brotherhood of trying to monopolize power while neglecting the country’s economic crisis. But the opposition has struggled to capitalize on the dissent with a unified message and goal.

On Wednesday, the liberal NSF appeared to shift its strategy when it joined with its former foes in the ultraconservative Salafist Nour party to call for a national dialogue, days after it had rejected a similar call from Morsi.

But Morsi’s administration, while hailing the dialogue efforts, has largely dismissed other opposition demands, including calls for a new cabinet, constitutional amendments and early presidential elections. On Friday, many members of the opposition expressed frustration about a protest movement that has resulted in more divisions than concessions.

“The battle was pretty intense the past few days, and a lot of people feel tired and disheartened,” said Ahmed Ali, a doctor who joined the crowd outside the palace on Friday afternoon, before clashes erupted. “Let’s fool ourselves into thinking that a lot of people didn’t come today because of the weather.”

Ingy Hassieb contributed to this report.