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Fighters opposed to the government in Moscow say they have have captured two Russian soldiers in Belgorod, near the border with Ukraine.
Belgorod's top official said he had agreed to meet the men's captors if the soldiers were still alive.
Russia has blamed Ukraine for recent attacks in its border territories, but Kyiv denies being directly involved.
The Russian army said on Sunday its artillery had hit a "terrorist" group near the village of Novaya Tavolzhanka.
"The enemy scattered and retreated," the statement added.
Earlier, a group of paramilitaries issued a message on the Telegram app, saying they had captured the two men but would hand them over if Belgorod governor Vyacheslav Gladkov agreed to talks.
The video purported to show both captives, although the BBC has been unable to independently verify their identities.
The message was posted by the Russian Volunteer Corps (RDK), and described as a joint statement with the Liberty of Russia Legion (FRL).
Both groups want to topple President Vladimir Putin, and also oppose the full-scale invasion of Ukraine that he launched in February last year.
Ukrainian officials say the two paramilitary organisations consist of Russian citizens who want to create a "security zone" for Ukrainians.
The RDK came to prominence in March for a cross-border raid in Russia's Bryansk region. Its leader is a Russian nationalist with alleged links to neo-Nazis.
The FRL is considered a different sort of organisation that fights alongside Ukrainian troops against Russian forces.
Mr Gladkov replied with his own Telegram video, saying he had agreed to the talks if the soldiers were shown to be still alive, adding that he thought they had probably been killed.
He labelled the fighters in question "scoundrels, murderers, fascists", but promised to "guarantee safety" if the talks took place.
And though they asked him to go to Novaya Tavolzhanka to meet them, he said this was too dangerous and that he would expect them at a checkpoint in the town of Shebekino.
It was not immediately clear whether any head-to-head had taken place. Mr Gladkov also said fighting with a group of "Ukrainian saboteurs" was taking place.
Kyiv has denied having any direct involvement in such attacks.
But it has painted the growing violence in Russian territory as being the inevitable consequence of Russia's invasion last year.