"Communication is still difficult. We will find out today how the fighting ended yesterday. We still feel we're still sharing Malakal with the rebels, but by the end of the day (the army) must clear the rebels from Malakal," army spokesman Philip Aguer told AFP.
Both rebels and the government claim to control the northeastern town, the capital of Upper Nile state and one of the main battlefields since fighting erupted last month between rival forces loyal to President Salva Kiir and his sacked deputy, Riek Machar.
Aguer said fighting also continued today over two other strategic regional capitals, Bor in the eastern state of Jonglei and Bentiu in northern Unity state.
The United Nations said yesterday that child soldiers had joined the conflict, with atrocities including war crimes committed on both sides.
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"The reports that we have come across involve mass killings, extrajudicial killings, widespread destruction and looting, and child soldier recruitment," UN Assistant Secretary General for Human Rights Ivan Simonovic told reporters as he wrapped up a visit to the country.
The United Nations says 468,000 people have fled their homes because of the fighting, which has spiralled into ethnic killings between members of Kiir's Dinka people -- the country's largest group -- and Machar's Nuer.
Up to 10,000 people are believed to have been killed so far in the conflict.