* Platon Lebedev still faces tax claim after release
* Tax issue means Khodorkovsky likely to stay abroad
* Putin wants to polish Russia's image before Olympics
By Maria Tsvetkova
MOSCOW, Jan 23 (Reuters) - Russia's Supreme Court ordered the release of Kremlin critic Mikhail Khodorkovsky's business partner on Thursday after a decade in jail, but refused to drop a tax claim that is keeping the former oil tycoon abroad.
Platon Lebedev is the latest high-profile prisoner to be freed in Russia as President Vladimir Putin tries to soften criticism of his record on human rights and democracy before next month's Winter Olympics in the Black Sea resort of Sochi.
But the decision not to lift a demand that the pair pay the state 17 billion roubles ($521 million) in tax arrears reduces the likelihood of Khodorkovsky returning to Russia from Germany and challenging Putin.
"Concerning the Supreme Court's decision to overturn the tax claim: I get the message," Khodorkovsky said in a telephone interview with the New Times, an independent news weekly. "It can mean only one thing: I am not welcome in Russia."
The jailing of Lebedev and Khodorkovsky for tax evasion and fraud in 2005 was widely seen abroad as politically motivated after Khodorkovsky, once Russia's richest man, was arrested at gunpoint on an airport runway in 2003.
Khodorkovsky was pardoned by Putin in December but Lebedev, 57, refused to seek clemency and remained in jail near the Arctic Circle. He had been due for release in May but the Supreme Court reduced his sentence to time served.
"Release Lebedev, Platon Leonidovich, due to the completion of his punishment," Supreme Court Judge Pyotr Serkov declared after reading out his ruling, for which Lebedev was not present.
Lebedev's lawyer, Vladimir Krasnov, welcomed the fact that his client would be able to walk free on Friday but regretted that the court had not overturned his conviction.
Khodorkovsky's legal team said it would fight the tax fine. Shortly after his release and flight to Germany in December to be with his sick mother, Khodorkovsky cited the need to pay the tax arrears as an obstacle to returning to Russia.
"The main thing today is the release of Platon Lebedev, who became one of the first hostages in the Yukos case in 2003," Khodorkovsky's lawyers said in a statement on his website.
But they said "the (tax) claim deprives Mikhail ... of an opportunity to come back to Russia because as long as the claim stands, the 'iron curtain' can drop in front of him any time."
Rights groups viewed Khodorkovsky and Lebedev as political prisoners, and Khodorkovsky's lawyers described the case as "one of the most shameful pages" of Russia's post-Soviet history.
Yukos was broken up and sold, mainly into state hands. Its prize production asset went to state oil company Rosneft , which is now headed by close Putin ally Igor Sechin.
The Kremlin denies using the courts for political purposes and did not comment on the Supreme Court's decision, taken after a recommendation by the European Court of Human Rights that Russia should review the sentencing in the Yukos case.
The Strasbourg-based court ruled last July that the first case against Khodorkovsky and Lebedev was legitimate but found that the trial was unfair and the sentencing unjustified.
In the eyes of critics at home and abroad, the treatment of Khodorkovsky and, to a lesser extent, Lebedev was a significant stain on the record of Putin, who is now serving his third term as president after a stint from 2008 to 2012 as prime minister.
Putin has sought to address international concerns over what critics see as a crackdown on opponents in his new term as he prepares to host the Sochi Olympics, on which he has staked much of his personal and political prestige.
Prominent inmates freed in the past few weeks, as part of a broader amnesty, also included two women from the Pussy Riot protest group jailed over their performance of a "punk prayer" critical of Putin in a Russian Orthodox church in Moscow.
The amnesty also ensured a group of activists from the environmental group Greenpeace did not go on trial over a protest at an Arctic oil rig, a case in which Russia's threat of seven-year jail terms was widely seen abroad as excessive.