The anti-Kremlin Freedom of Russia Legion claimed responsibility for the infiltration of Russian territory.
Russia said Tuesday that its armed forces had beaten back and then wiped out a group of militants and military hardware that were dispatched to its southern Belgorod region from Ukraine.
The incursion was the most serious attack on Russian soil since the beginning of Moscow's offensive in Ukraine, prompting Russian authorities to declare an "anti-terror regime" and evacuate nine border villages.
The announcement of the incursion came after Kyiv said Russian forces pummelled the eastern Ukrainian city of Dnipro with missiles and drones overnight.
The Kremlin said that President Vladimir Putin had been informed of the cross-border incursion, adding Moscow believed the attack was designed to "divert attention" from the situation in Bakhmut.
Vyacheslav Gladkov, the governor of the southern Belgorod region, which has been targeted by attacks since the beginning of Russia's invasion of Ukraine, said troops and members of the security service were fighting a "sabotage" group that crossed over from Ukraine.
Gladkov said that troops and members of the FSB security service were "taking the necessary measures to eliminate the enemy."
Members of the Freedom of Russia Legion claimed responsibility for the attack.
In a video released by a Telegram channel claiming to represent the group, a camouflaged spokesman, surrounded by armed men in fatigues, said: "Russia will be free!"
The group's Telegram channel said that two villages in the Belgorod region had been attacked.
Kyiv denied involvement.
"Ukraine is watching the events in Russia's Belgorod region with interest and is studying the situation, but has nothing to do with it," Ukrainian presidential advisor Mikhaylo Podolyak said.
After Putin sent troops to Ukraine in February 2022, the border region of Belgorod has been repeatedly shelled, and dozens of people have been killed.
Authorities have also reported a series of incidents involving members of "sabotage" groups crossing over from Ukraine.
In April, the city of Belgorod was bombed by a Russian fighter jet which dropped ammunition over the city by accident.
The attack was reported ahead of a widely-expected Ukrainian offensive, though President Volodymyr Zelensky has said his country is not yet ready.
Ukraine's army said earlier Monday that Russian forces pummelled Dnipro with 16 missiles and 20 attack drones overnight.
The army said its air defence forces had destroyed all the drones and four of the cruise missiles.
Both the Russian mercenary group Wagner and the regular army said over the weekend that Bakhmut had fallen following months of fighting, but Ukraine on Monday said its troops continued to fight for the devastated city.
Kyiv said it still controlled a small area of the city.
"The fighting continues," Ukraine's Deputy Defence Minister Ganna Malyar said, a day after Zelensky said Bakhmut was "not occupied" by Russia.
"The battle for the dominant heights on the flanks -- north and south of the suburbs -- continues," she added.
On Monday, Wagner chief Yevgeny Prigozhin, whose forces have spearheaded the grinding, months-long Russian assault on Bakhmut, said the mercenaries would leave the city by June 1 and hand over control to regular Russian troops.
"Wagner will leave Artemovsk from May 25 to June 1," Prigozhin said in an audio recording on Telegram, referring to the city by its Russian name.
Prigozhin said the mercenaries had set up "defence lines" on the city's western outskirts before a planned transfer of control to the Russian army.
"If the ministry of defence does not have enough personnel, we have thousands of generals," he said.
Prigozhin has scathingly criticised Russian Defence Minister Sergei Shoigu and Valery Gerasimov, Russia's chief of general staff, for blunders in Ukraine, accusing them of being incompetent and causing large-scale losses.
Nuclear plant power cut
Following attacks on the region of Dnipro, Ukrainian authorities said that the occupied Zaporizhzhia nuclear power plant in the southeast went offline for several hours.
Ukraine's nuclear agency Energoatom said it was the seventh time the plant entered "blackout mode" since Moscow's troops took control in March 2022.
"The station is switching to power from the Ukrainian energy system," Ukrenergo, Ukraine's state grid operator, said later Monday.
The operator said that despite the morning attack Ukraine's energy system was operating in a "stable fashion" and that there was enough electricity to "cover the needs of consumers."
Located in the southeastern region of Zaporizhzhia, the six-reactor plant is the largest in Europe.
The grid operator said that in the eastern region of Dnipro, high-voltage lines had been damaged as a result of Moscow's latest strikes.
The last power cut at Zaporizhzhia had been caused by another wave of Russian missile attacks, Energoatom said.
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