Giza, Egypt (CNN) -- Egyptians headed to the polls Wednesday for a critical round of voting to shape the new parliament and help determine how much power Islamists will hold in building the country's next government.
The second round in the country's three-stage election process, Wednesday's vote is vital because it includes the Giza, Luxor, Aswan and Ismailia regions, which have historically favored conservative Islamic candidates.
Officials at polling stations reported what appeared to be good turnout. But one instance of violence was reported.
State-owned egynews.net reported that army personnel stopped a fight between two voters by shooting their guns in the air at a polling station in the area of Saff, near Cairo. Voting continued shortly after, the report said.
Egypt's Islamists claimed victory in the first round of elections in other parts of the country last week.
Secular parties stepped up their campaigning following those early results. The Revolution Continues coalition increased grass-roots efforts, and the Egyptian Bloc used the slogan: "For a more balanced parliament."
"It became clear to the secular parties that they were spoiling each other's chances," Egyptian political analyst Hisham Kassem said Wednesday. "So instead of competing against each other, they decided to help each other against the Islamists."
Their joint efforts "could improve their chances slightly," he said. "Ultimately, however, the most important part of these elections is that it is giving the Egyptians an electoral map of Egypt, which did not exist before."
Early results from the first round of voting triggered tensions between leaders of the Islamist parties and the military council that is running Egypt until a new government is formed. A popular uprising ousted President Hosni Mubarak in February.
Lt. Col. Amr Imam, a spokesman for the Supreme Council of the Armed Forces, said last week, "We will not allow a dictatorship and we do not want to remain in power, but we do want a civil government representing all Egyptians without marginalizing any minorities."
He complained the first-round results did not "represent all factions of the society, like the Coptics for example," in reference to the country's largest Christian sect.
According to media reports, Gen. Mukhtar al-Mulla, a high-ranking military official, said the military would increase its role in the drafting of Egypt's constitution.
The Muslim Brotherhood, the biggest Islamist party, assailed those remarks, calling them "almost humorous and leaning towards calling for a new dictatorship."
The deputy director of the Muslim Brotherhood's party said it would pull out of participating in a government advisory board.
Shibley Telhami, a professor at the University of Maryland and senior fellow at the Brookings Institution, said the military leaders' remarks "played into the hands of those who worry that the military institution does not intend to relinquish political control."
The results of Wednesday's elections will help indicate whether the tensions are impacting votes, and will give an indication of how much strength minority parties are likely to have in parliament.
Egypt's official State Information Service said 33 parties were fielding 3,387 candidates to compete for 180 seats in this round of the elections. Some are individual seats, others are seats given to party lists.
The Muslim Brotherhood's Freedom and Justice Party, which is relatively moderate, has said on its website that it won 34 seats after the initial round of voting.
The conservative Al Nour Salafi Party won five seats, according to its website.
The Muslim Brotherhood has pledged to respect minority rights and work with liberal parties, while al-Nour supports broader application of its interpretation of strict Islamic law in Egypt.
A liberal voter CNN spoke with in Giza complained Wednesday about suggestions by a leader of the al-Nour party that alcohol and bikini bathing suits should be banned. Jaylana Missiri said that's not what the government should be focusing on. "We should be talking about people who are living in poor neighborhoods and poor conditions and how we can better their lives," she said.
These are the first parliamentary elections since Mubarak was forced from office.
Many voters believe that for the first time in their lives, their votes matter in this election.
At one polling site, about 100 people waited in line to cast their ballots Wednesday.
Abdul Kareem Mohammed said the first round of the election made him feel proud about the changes in his nation.
"I am happy to vote for the first time in my life today," Mohammed, 28, said. "This is the voice of the Egyptian people."
Voter Hani Salah al-dine said he voted for the Muslim Brotherhood. He said he believed the first round of the election was fair, citing the outcome of obscure candidates who did better than some rich, well-known candidates.
"This show's the common man's voice is being heard," he said.
CNN's Josh Levs, Mary Rogers, and Raja Razek contributed to this report