UPDATE 2-Russia relaxes 2010 gas demands on Ukraine

* Gazprom, Naftogaz agree to reduced gas volumes

* Ukraine to import 33.75 bcm of Russian gas in 2010

* Gazprom agrees to waive fines for 2009 (Adds details, background)

By Robin Paxton and Pavel Polityuk

MOSCOW/KIEV, Nov 24 (Reuters) - Russia relaxed demands on Ukraine for importing gas in 2010 and agreed on Tuesday to waive fines on this year's supplies in a deal easing European fears of a renewed dispute along a route supplying a fifth of its gas.

Gazprom (GAZP.MM), Russia's powerful state-run gas export monopoly, agreed that Ukraine -- whose economy has been crippled by the global financial crisis -- could buy 35 percent less gas than originally contracted for next year.

"The volumes have been corrected in accordance with actual consumption in Ukraine in conditions of crisis," Gazprom Chief Executive Alexei Miller said after four hours of talks with Oleh Dubyna, the head of Ukrainian state energy firm Naftogaz.

"These volumes remove the risk of the payment of fines by Naftogaz Ukraine in 2010 for the non-use of contracted gas." Russian gas to Europe via Ukraine was cut off for three weeks in January after the ex-Soviet states argued over pricing and transit, a dispute that shook confidence in Russia's ability to supply gas and Ukraine's ability to give it safe passage.

Europe is concerned the dispute, which damaged industry and left hundreds of thousands in the cold, could flare up again at New Year, especially as rival political factions in Ukraine jostle for support ahead of a Jan. 17 presidential election.