Ukraine's Supreme Court is considering claims of massive fraud in the presidential elections, amid calls to prevent the country breaking up.
Rival supporters are confronting each other on Kiev's streets
The court has begun hearing opposition candidate Viktor Yushchenko's appeal against the official poll result.
Supporters of his rival, Prime Minister Viktor Yanukovych, have said they will hold a referendum on autonomy if Mr Yushchenko is declared winner.
But the court has now said it will not issue any ruling on Monday.
Outgoing President Leonid Kuchma said on Monday he could not accept any division of Ukraine, after talks with regional leaders who have threatened to demand autonomy.
"My position is that we cannot allow the division of Ukraine," Mr Kuchma said in televised remarks.
UKRAINE SUPREME COURT
Considering Yushchenko camp's complaints of poll abuses and arguments of pro-government camp
Cannot order a new election, but can annul results in specific districts, paving way for a re-run
At least 22 judges meeting on Monday
Upheld some opposition complaints after first round
His warning was echoed by EU foreign policy chief Javier Solana, who said "the unity of Ukraine is fundamental".
Speaking in Brussels after a meeting with EU and Nato ambassadors, he said dialogue must continue in Ukraine and "what we have to avoid is the outbreak of violence".
Nato Secretary-General Jaap de Hoop Scheffer agreed that maintaining Ukraine's territorial integrity was essential.
"The sense of belonging to one nation is very important and on that basis a solution should be found," he said.
Fears of a split in Ukraine intensified after the eastern Donetsk region on Sunday scheduled a referendum on autonomy.
Officials from some other regions also met to consider seeking autonomy in case Mr Yushchenko - a pro-Western liberal - becomes president.
The parliament of Donetsk region will hold an emergency session on Tuesday to consider the plan to hold a referendum on 5 December.
POLITICAL CRISIS TIMELINE
21 Nov: Viktor Yanukovych declared winner of run-off poll
Independent observers declare the elections flawed, and thousands take to the streets
25 Nov: Supreme Court suspends publication of result until it considers the opposition's complaints
26 Nov: Mr Yanukovych and Mr Yushchenko hold talks and agree to seek peaceful solution
27 Nov: MPs declare election invalid, pass vote of no-confidence in the election commission
28 Nov: Eastern regions threaten to secede if Mr Yushchenko is declared president
Focus on court
Near the court building in central Kiev, several thousand rival supporters stood together, shouting "Yu-shchen-ko!" and "Yanu-ko-vych!" and waving the orange flags of the opposition and blue-and-white Yanukovych flags.
Last week the court suspended the official results, which saw Mr Yanukovych, who is seen as pro-Russian, declared the winner with 49.46% of the vote to Mr Yushchenko's 46.61%.
Mr Yushchenko's team has submitted thousands of allegations of ballot-rigging to regional courts and one complaint to the Supreme Court.
But the government has also lodged 7,000 complaints of irregularities to regional courts.
Mr Yushchenko has declared himself the rightful winner, alleging massive fraud. His concerns have been backed by international observers.
The Russian government has suggested it might reverse its opposition to a new election. The EU has already called for a re-run.
But the Supreme Court could take up to a week to reach a decision.
The court - seen as relatively independent - cannot invalidate the whole election, but it can uphold a complaint and order a partial or full recount.
It could on the other hand proclaim Mr Yanukovych the winner.
The BBC's Jonathan Charles in Donetsk says the referendum warning and the more measured response of other eastern areas is clearly meant to be a threat - a way of putting pressure on the crisis talks.
Addressing his supporters earlier, Mr Yushchenko made it clear he would not tolerate a break-up of the country.
"Those people who will raise the issue of separatism will be held criminally responsible under the Ukrainian constitution," he said.
Meanwhile, the head of Mr Yanukovych's campaign team, Serhiy Tyhypko - who also headed the Ukrainian central bank - has resigned from both posts.
Mr Tyhypko said Mr Yanukovych had been badly advised to support moves in the Donbass towards a referendum on autonomy. He said talk of splitting Ukraine was "madness".