Egyptian protesters clashed with security forces on Friday near the presidential palace in Cairo, where hundreds held a rally against Islamist President Mohammed Morsi, DPA reported.
Security forces then set on fire tents erected by protesters around the palace.
The presidency said in a statement that security forces would deal "with the utmost decisiveness to enforce the law and protect state institutions."
Media reports said the violence was started by masked men, whose affiliations could not be confirmed.
Thousands took to the streets on Friday after a week of deadly clashes, demanding that Morsi form a national unity government.
His Muslim Brotherhood group called on all opposition forces to withdraw from around the palace.
"There is no time for condemnation, but there must be action on the ground to prevent any political cover for acts of violence, which we all denounce," the group said.
In central Cairo, hundreds chanted "down with the Brotherhood" as they marched to Tahrir Square, despite heavy rain.
In the Delta city of Kafr al-Sheikh, 18 people were injured in clashes between protesters and police, state media reported.
However, the biggest march was in the city of Port Said, which saw the worst of the violence, with thousands calling for retribution.
They chanted "The people want to bring down the regime" and "Down with the Supreme Guide's rule," referring to the religious head of the Muslim Brotherhood.
Some of the protesters demanded Morsi's overthrow, chanting "Leave, Leave."
Clashes began on the eve of the second anniversary of the January 25 revolt that ousted Hosny Mubarak. They were given fresh impetus when an Egyptian court handed down death sentences to 21 Port Said locals for their role in a deadly riot at a football match last year.
Friday's protests coincide with the first anniversary of the football match, at which 74 people were killed. Protesters said the death sentences were politically motivated.
The Health Ministry said five people died Friday in Port Said of injuries sustained during previous clashes and one was killed in Cairo.
The new deaths bring to around 60 the death toll since the protests began last week.
Islamists and the opposition have traded blame for the violence.
The opposition accuses Morsi, the country's first democratically elected president, of tightening the Brotherhood's hold on power and failing to revitalize an ailing economy.
In Egypt's second city of Alexandria, thousands of protesters marched to the city centre, amid tight security around government buildings.
Hundreds more held rallies in the cities of Zaqaziq and al-Mahalla. In Suez, protesters carried symbolic coffins to mourn those killed in the violence.
"We brought down the Mubarak regime through a peaceful revolution and we insist on achieving our goals by the same method, regardless of what the sacrifices or how savage the oppression would be," opposition leader Mohammed ElBaradei said.
"Our strength is in our unity, numbers and peacefulness."
ElBaradei, one of the founders of the National Salvation Front coalition, renewed the demands by the alliance for a new government, amending the constitution and ensuring the judiciary's independence.