* Far-right radical group protests at parliament
* Acting president: Russia wants to destabilise Kiev
* Ukrainian capital has 526 defence shelters
By Richard Balmforth
KIEV, March 28 (Reuters) - Ukrainian authorities carried out an inspection tour of Kiev's bomb shelters on Friday as lawmakers accused Russia of fomenting trouble in the Ukrainian capital after having annexed the Black Sea region of Crimea.
The new leadership seemed on edge after a far-right radical group that played a central role in the revolt that overthrew president Viktor Yanukovich rallied angrily outside parliament, demanding the dismissal of interior minister Arsen Avakov.
"They (the Russians) did not manage to ignite the flame of separatism in our regions," said acting president Oleksander Turchinov. "So now there are attempts to destabilise the situation in the heart of Ukraine, in Kiev."
Tens of thousands of Russian troops thought to be massed on the border have shown no immediate sign of entering other parts of Ukraine, and President Vladimir Putin has said Moscow has no designs on other parts of its smaller neighbour. But Russia is expected to use every weapon in its economic arsenal to punish Ukraine for its U-turn towards Europe.
In an unusual step that added to a climate of apprehension, Kiev authorities announced they had carried out checks on more than 500 urban bomb shelters in the capital to ensure they were in good working order, and were working on a new early warning system for the population of the former Soviet republic.
"We have 526 defence installations (shelters) in Kiev ... Today the city authorities are working to ensure that they are in appropriate technical condition to be able to guarantee the protection of people," Volodymyr Bondarenko, head of the capital's administration, said in a statement.
Ukraine's new leaders, who took power after the pro-Russian Yanukovich fled on Feb. 20 following three months of sometimes violent unrest, appeared unnerved after Right Sector, an ultra-radical group, staged a protest outside parliament on Thursday night and threatened similar action on Friday.
The group came to prominence during the three-month "Euromaidan" revolt against Yanukovich by breaking away from the largely peaceful demonstration to use violent tactics against riot police, throwing petrol bombs and bricks.
Although unable at the time to condone the violent tactics used by the group, the then-opposition leaders who are now in power could not begrudge the part Right Sector played in eventually toppling Yanukovich.
ACTIVIST'S KILLING ANGERS FAR RIGHT
However, Turchinov suggested on Friday that the group might be manipulated by Moscow into undermining the new leadership and destabilising the country as it reels from Russia's seizure of Crimea and seeks to recover from economic mismanagement.
The stated reason for the Right Sector protest was the killing earlier this week of Oleksander Muzychko, also known as Sashko Bily, one of the group's most prominent activists. ID:nL5N0MM26Q]
The interior ministry said Muzychko was shot dead by officers of the "Sokol" special unit as he tried to escape from a cafe in the western region of Rivne. The ministry modified its statement on Friday, saying ballistics tests showed that Muzychko had shot himself as police closed in.
Right Sector says Muzychko was the victim of a political "hit" by special forces and is demanding the resignation of Avakov and the prosecution of those responsible for the killing.
Turchinov pledged a thorough investigation into Muzychko's killing would be thoroughly investigated, but urged Ukrainians to show solidarity with the interim leadership.
"Unfortunately, yesterday certain citizens consciously or unconsciously attempted to carry out a 'provocation' outside parliament," Turchinov said. If people did not approve of what parliament was doing, they had recourse to elections, he said.
"The other course, the use of force and pressure and provocations, is a course that leads to catastrophe for Ukraine.
"This is a course that is against our state and supports the aggressor which is now concentrating its armed forces on the border of Ukraine," he said.
(Additional reporting by Natalia Zinets; Editing by Mark Heinrich)