Voters casting ballots on Tuesday will decide the outcome in 133 seats where no candidate scored more than half the vote in last week's polling, the first of the three-round elections for the People's Assembly.
Hussain Abd al-Ghani, Aljazeera's Egypt bureau chief, reported that the competition between the ruling party and the Muslim Brotherhood is fierce.
The rivalry takes on a sectarian tone - setting Muslim against Coptic candidates - in some constituencies.
Fifteen candidates from the ruling National Democratic Party (NDP) were contesting the southern Egyptian constituency of Minya against six Muslim Brotherhood candidates, said Aljazeera correspondent Amr al-Kahki.
But the fiercest competition was between Muslim Brotherhood candidate Muhammad Saad al-Katatni and the Marxist Tagammu party's Wajih Shukri, he said.
Voter turnout in Minya was very low on Tuesday morning.
Some complaints were reported by representatives of the Muslim Brotherhood candidates in the three Upper - or southern - Egyptian governorates of Minya, Asyut and Bani Suwaif, al-Kahki reported.
A representative of the Brotherhood candidate in Asyut, Khalid Awda, told Aljazeera that a group of rioters had attacked Wahid Muhammad Hassan and another representative in western Asyut, beating them with sticks.
The Muslim Brotherhood has
shown strength at the ballot box
In Bani Suwaif, fighting broke out at the entrances of some electoral centres, and some voters in the villages of al-Tawfiqiya and al-Hawarta in Minya were prevented from casting ballots, Aljazeera's al-Kahki said.
In 9 November's first round of voting, influential members from the NDP sought re-election amid reports of widespread corruption and vote-rigging.
After the first round, there was no clear indication on the shape of the next parliament.
Complicating matters, the results announced in three of the 82 constituencies involved in the first round were annulled on Sunday and will lead to re-runs.
The NDP holds 404 out of 454 seats in parliament, and observers have predicted that opposition movements might make only small inroads into its domination.
The officially banned Muslim Brotherhood has capitalised on recent freedom to conduct an intensive and well-crafted campaign, which it claims should help treble its current seat tally of 15.
But the first round was marked by the failure of the secular opposition to organise a challenge.
Al-Ghad party leader Ayman Nour, who came second in the presidential election behind Mubarak, lost in his own Cairo stronghold, but filed a complaint alleging vote rigging.
A coalition including the Kefaya (Enough) movement, the Marxist Tagammu, the Nasserists and the liberal Wafd failed to secure a seat in the first round.
Independent monitors, opposition candidates and judges said that besides the corruption that characterised parliamentary campaigns, irregularities included falsified voter registries and ballot-stuffing during the counting process.
The elections were planned in three two-round phases. The second, which includes Alexandria, is to begin on 20 November, and all 26 governorates in the country will have finished voting by 7 December.