Russian troops are expected to pull out of Gori and other Georgian towns
There is disagreement between Moscow and Tbilisi over whether or not Russia has begun withdrawing troops from Georgia, as promised.
Announcing a "pull-back" from Georgia proper into South Ossetia, a Russian general made clear he did not regard the province as Georgian territory.
According to Russian media, some combat units inside South Ossetia were withdrawn to Russia during the day.
A top Georgian official said there was no evidence of any Russian withdrawal.
Russian President Dmitry Medvedev earlier promised to abide by the terms of a ceasefire brokered by French President Nicolas Sarkozy.
We are talking about pulling back to the territory of South Ossetia
Col-Gen Anatoly Nogovitsyn deputy chief of the Russian General Staff
The conflict over South Ossetia erupted 11 days ago, when the Georgian army tried to wrest back control and Russia sent in its troops to drive them out.
Russian troops remain stationed near the Georgian capital Tbilisi, with Moscow saying it has the right to keep some troops as peacekeepers in a buffer zone around South Ossetia.
The West has repeatedly urged Russia to withdraw its troops and France has said it will call a special EU summit if Russia does not comply.
Mr Medvedev, who was in North Ossetia to decorate Russian soldiers for valour during the recent fighting, has promised to "do whatever is necessary" to maintain security in the region.
"If anyone thinks he can kill our citizens without being punished, or kill our soldiers and officers, who are peacekeepers, we will never allow this," he said.
He added that Russian soldiers had demonstrated that they had recovered from the crisis of the 1990s and were now a fighting force again.
'Pulling back - not withdrawing'
"The pull-back of peacekeeping forces started today," the deputy chief of the Russian General Staff, Col-Gen Anatoly Nogovitsyn, said in Moscow.
Georgian TV has shown video of what it says is Russian armour pushing aside police cars in Igoeti, around 48km (30 miles) from Tbilisi, on 18 August
"We are talking about pulling back to the territory of South Ossetia. There should be no troops on the territory of Georgia."
Gen Nogovitsyn said that President Medvedev had agreed with President Sarkozy by telephone on Sunday on a "pull-back, not a withdrawal" of Russian troops.
Russian news agencies reported that several Russian military units, each mounted in five to 10 vehicles, left the South Ossetian capital Tskhinvali on Monday for North Ossetia. The reports could not be confirmed independently.
A spokesman for the Russian defence ministry, Lt-Gen Nikolai Uvarov, told BBC News that troops were being withdrawn from Georgia and the process would take "days not weeks".
Confirming that Russia would retain its pre-conflict peacekeepers in South Ossetia, Gen Uvarov suggested that they would also be beefed up.
"We used to have there one light infantry battalion, less than 600 people, certainly it will be more soldiers but not just soldiers... [the] force must be credible to prevent any further escalation of conflict there," he said.
No move north
Alexander Lomaia, secretary of the Georgian National Security Council, told the BBC he had seen no evidence of Russian troops leaving Georgian soil on Monday.
No more use of force
Stop all military actions for good
Free access to humanitarian aid
Georgian troops return to their places of permanent deployment
Russian troops to return to pre-conflict positions
International talks about security in South Ossetia and Abkhazia
As of Monday evening, there was no sign of a withdrawal on the highway from the Georgian capital Tbilisi to Gori, a town close to South Ossetia which Russian forces now control, the BBC's Gabriel Gatehouse reports.
The BBC observed five Russian armoured personnel carriers, each with around a dozen men, heading in the direction of the capital.
However, there was no suggestion that they ventured any further than the village of Igoeti, where the furthest Russian checkpoint is located.
Military hostilities may have ended, our correspondent adds, but the diplomatic war looks set to continue.
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