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MOSCOW, May 27 (Reuters) - Russian President Vladimir Putin called on Tuesday for an immediate halt to Ukraine's military operation against pro-Russian separatists in the east of the country.

In his first reported comments on Ukraine since its presidential election on Sunday, Putin also stepped up pressure on Kiev to start a dialogue with the rebel leaders as fighting raged in east Ukraine.

The Kremlin said Putin had spoken to Italian Prime Minister Matteo Renzi by telephone and "underscored the need for an immediate halt to the punitive military operation in the southeastern regions and the establishment of peaceful dialogue between Kiev and representatives of the regions."

Moscow has amassed troops on the frontier with Ukraine during the crisis, in which Ukraine's Moscow-basked president was ousted and Russia annexed the Crimea region from Ukraine, but said last week they had started to withdraw.

NATO has noted, however, that many still remain.

Foreign Minister Sergei Lavrov said separately that halting bloodshed was "the No. 1 task for the Kiev authorities and a test of their durability" after billionaire businessman Petro Poroshenko's election success.

"The election has taken place, and we respect the results of the expression of the people's will, but we believe it is absolutely imperative to ... stop all violence," Itar-Tass news agency quoted him as telling a news conference.

Lavrov said Poroshenko would find Russia a reliable partner if he established dialogue with eastern Ukraine, but said that no visit by Poroshenko to Russia was being considered.

He also warned the government that efforts to put down the separatists could backfire.

"If they are counting on ... suppressing resistance in the southeast before Poroshenko's expected inauguration and enabling him to go there as a victor, I don't think this will create good conditions for a warm welcome in the Donetsk region (in eastern Ukraine)," Itar-Tass quoted Lavrov as saying.

(Writing by Steve Gutterman, Editing by Timothy Heritage)

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