* Parliament also votes no-confidence in prime minister
* Putin aide held talks with political leaders
* Russian-backed region broke from Georgian control in 1990s
By Steve Gutterman
MOSCOW, May 29 (Reuters) - Parliament in Georgia's Russian-backed breakaway Abkhazia region called on the president to step down on Thursday after three days of unrest during which opponents stormed his headquarters following a protest over alleged corruption and misrule.
Lawmakers adopted a declaration urging President Alexander Ankvab to resign and voted no confidence in the prime minister of the Black Sea province, which Russia recognised as an independent nation after Moscow's 2008 war with Georgia.
There was no immediate response from Ankvab after the vote, taken by a show of hands during a late-evening session broadcast in part on Russian TV. It was backed by 20 of the 21 lawmakers present, out of a total of 35, Russian news agencies reported.
"If Alexander Ankvab resigns voluntarily, we are ready to discuss all issues linked to his security and guarantees for the future," Interfax quoted an opposition leader and former prime minister, Sergei Shamba, as saying.
"There will be no witch hunt. What is important today is to liquidate the rift in Abkhaz society and unite the people."
Ankvab left the presidential headquarters after protesters demanding reforms and an end to corruption broke windows and doors and entered his headquarters on Tuesday while he was holding inconclusive talks with opposition leaders.
He said in a statement on Wednesday that he was still in the country and was looking for ways to resolve the crisis.
The voting came after a close aide to Russian President Vladimir Putin, Vladislav Surkov, travelled to Abkhazia along with a former Russian interior minister and held talks with Ankvab and opposition leaders, according to Russian media.
Surkov seemed to suggest that unrest was not in the interests of Abkhazia and could jeopardize Russian subsidies.
He said the purpose of his visit was "to discuss implementation of the socio-economic development plan for Abkhazia in the conditions of high political tension," Interfax reported. "This plan is financed entirely by Russia."
Bloodshed or protracted unrest in Abkhazia could be embarrassing for Russia, which recognised the region and another breakaway province of Georgia, South Ossetia, as independent states after fighting a five-day war against Georgia in 2008.
Abkhazia broke from Georgian rule in a 1992-1993 war after the Soviet collapse.
Russia increased its control over Abkhazia after its own five-day war with Georgia and Russian forces help guard the region's de facto border with Georgian government-controlled territory.
Abkhazia's dependence on Russia is a source of political tension.
"The country cannot continue to go with the current, relying only on subsidies from friendly Russia," Interfax quoted a declaration adopted by opposition demonstrators outside the presidential administration building on Thursday as saying. (Writing by Steve Gutterman; editing by Stephen Addison)