Lukashenko's office declined to comment on the state of his health while Russia denied he was ill.
The group said it was entering politics as an opposition party without any formal or informal instruction from the Kremlin.
As part of its economic response to Russia's invasion, Kyiv has imposed sanctions on Russia.
Kaliningrad, which sits in an exclave that is sandwiched between Lithuania and Poland on the Baltic coast, was known by the German name of Koenigsberg until after World War II.
It is the third bombing attack on Russian war supporters, the Russian foreign ministry blamed Kyiv and the West.
Moscow has been very concerned about protecting the Kremlin from drones since at least 2015 when it began using electronic countermeasures to automatically direct them away by "spoofing" GPS location
White House national security spokesman John Kirby called the claims "obviously ludicrous."
The Kremlin said two drones had been used in the alleged attack on Putin's residence in the Kremlin citadel, and Ukraine declined responsibility.
The Kremlin has repeatedly dismissed speculation that Putin, Russia's paramount leader since 1999, is ill.
Nikolai Peskov, son of Russian President Vladimir Putin's spokesman Dmitry Peskov, was born in 1990 and lived in Britain in the decade following the 1991 fall of the Soviet Union.
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