Intensity of fighting between pro-Russian separatists and government forces raises question whether truce will hold.
Ukraine’s government and pro-Russia rebels have said that a ceasefire that began at midnight local time (22:00 GMT) is holding but that there have been some violations.
The ceasefire is seen as a first step in a peace plan aimed at ending 10 months of conflict in eastern Ukraine.
Within two hours into the ceasefire, which came into effect on Saturday at midnight local time, Kiev reported attacks by separatists in the Luhansk area.
“Pro-Russian militants used artillery, mortars and grenade launchers to attack the Ukrainian positions … near Zolote, Sanzharivka, Popasna, Chornukhine by 2am,” said tweets by the Ukrainian army early on Sunday.
A pro-Kiev official said that two civilians were killed by rockets fired by rebels shortly after the start of the ceasefire, but most of the frontline had fallen silent.
An elderly man and woman died after Grad missile fire hit the town of Popasna in the Luhansk region about 20 minutes after the truce came into force, local governor Gennadiy Moskal said.
The firing reportedly came from an area that Kiev says is under the command of a renegade group of Cossack fighters who insist they will not obey the ceasefire.
Meanwhile, rebels accused the Ukrainians of deploying artillery shortly after midnight, the Associated Press reported.
Anxiety remains high that unrest could be sparked anew by rival claims to Debaltseve, a strategic railway hub now controlled by the government.
On Saturday, intense fighting persisted around Debaltseve, in the eastern part of Ukraine, which has been besieged by separatist forces.
Al Jazeera’s Charles Stratford, reporting from Donetsk, said at least seven Ukrainian soldiers were killed and 23 others wounded in Saturday’s violence.
Our correspondent said Sunday was a much quieter day.
“There has been a lot more confidence among the people in the street with a lot more people trying to conduct their everyday lives.
“We went down to the separatist-controlled town of Uglegorsk, which was quite close to Debeltseve, and indeed we did hear both incoming and outgoing shelling.
“There has been sporadic shelling in an around Debaltseve all day,” said Stratford.
He added that while a 14-day weapon’s withdrawal from the frontline, part of the ceasefire, was due to begin on Monday, the Observers from the Organisation for Security and Cooperation in Europe (OSCE) who have been tasked to monitor the truce, said they were prevented by the separatists from entering Debaltseve.
Under the terms of the ceasefire deal, approved on Thursday during talks between the leaders of Ukraine, Russia, Germany and France in Minsk, Belarus, the rival sides have two days from the start of the truce to start pulling back heavy weapons from the front line.
The peace plan is seen as the best hope of ending the violence that has claimed at least 5,480 lives since April, but scepticism remains high after the collapse of a similar previous deal.
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Minutes before the ceasefire, Ukrainian President Petro Poroshenko gave a speech where he ordered his military forces to abide by the agreement.
Military spokesman Vladyslav Selezynov said the Ukrainian armed forces immediately fulfilled Poroshenko’s order and the big guns fell silent in Donetsk and some other parts of the separatist-leaning east.
Hours before the ceasefire took effect, US President Barack Obama spoke to the leaders of Ukraine and Germany, during which he stressed the need for all sides to halt the violence.