Iraqi MPs assigned to investigate the killing of eight anti-government protesters by troops appealed for calm on a visit to the town on Sunday and pledged to publish their findings within days.
Their visit to the predominantly Sunni town 60 kilometres (35 miles) west of Baghdad came amid high tensions in the former insurgent bastion, where two soldiers were killed and three kidnapped on Saturday, a day after troops opened fire on demonstrators, killing eight.
The unrest came as lawmakers opposed to Prime Minister Nuri al-Maliki adopted a measure that would bar him from holding office beyond next year after weeks of angry rallies in mostly-Sunni areas against the Shiite premier's rule.
"We are sorry for what happened in Fallujah," said Shwan Mohammed Taha, a member of the Kurdistan Democratic Party and part of a parliamentary committee assigned to investigate the unrest.
"We hope that everyone calms down until the truth becomes clear."
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The shootings have dramatically ramped up tensions in Fallujah, with one Sunni Arab tribal leader giving authorities a one-week ultimatum to hand over the soldiers responsible for the killings.
Meanwhile, the head of the committee Khaled al-Attiya promised that the MPs would submit a report within 48 hours to parliament, after reviewing evidence and hearing witness accounts.
Friday's demonstration had been moving into Fallujah but was blocked by soldiers, who then opened fire after protesters began throwing bottles of water at them. Eight were killed and 59 wounded.
The following day, separate gun attacks on checkpoints and guard posts in Fallujah left two soldiers dead, one wounded, and three kidnapped.
The Fallujah demonstration was one of several across Sunni-majority areas of Iraq that have raged in recent weeks, hardening opposition against Maliki amid a political crisis ahead of provincial elections due in April.