Raila Odinga has said fresh elections may be needed
The Kenyan parliament's speaker has refused to rule on who should be leader of government business in parliament in the latest row to hit the coalition.
Kenneth Marende said the government must "consensually agree" who should hold the post.
Both Prime Minister Raila Odinga and Vice-President Kalonzo Musyoka say they must lead the House Business Committee.
It is the latest row between the PM's Orange Democratic Movement and the president's Party of National Unity.
The national assembly speaker's ruling was broadcast live on television on Tuesday.
The two parties agreed a power-sharing deal last year to end post-election violence, which had left some 1,500 people dead and forced 300,000 from their homes.
But relations have soured and crisis talks have failed to improve relations.
The president's allies say he has the power to decide who should hold the crucial position of leading government business in the house, but the prime minister's party disagrees.
The BBC's Josphat Makori in Nairobi says such a parliamentary dispute is unprecedented in the history of Kenya.
KENYA'S COALITION CRISIS
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5 Jan 2009:
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Odinga labels Kibaki's governing style "primitive"
Odinga tells Kibaki ODM will boycott cabinet meetings
Odinga says fresh elections may be needed
He says the ODM wanted to bring the contest to parliament in order to try to use its majority to have its way.
Our correspondent says this row matters because it has virtually paralysed the business of parliament.
Mr Musyoka used to be a senior ODM official but split to form his own party - ODM-Kenya - and is now seen as close to the president.
The prime minister recently complained that the vice-president should not be getting paid more than him.
Mr Odinga - who has claimed he is being sidelined in the unity government - has said fresh elections may be needed if the rift cannot be solved.
Over the weekend, President Kibaki's party accused the ODM of "fomenting a coup".
Earlier this month, Mr Odinga's party said it would boycott cabinet meetings, leading to inconclusive crisis talks.
The prime minister recently complained at a public rally that no red carpet or toilet were provided for him during an official visit.
The violence erupted after Mr Odinga's supporters said he had been cheated of victory in the December 2007 presidential election.
Former UN Secretary General Kofi Annan managed to broker a power-sharing deal in February 2008, which ended the violence.