UPDATE 3-Greece deal brings cheer to market battered by Italian politics

* Debt relief boosts Greek debt

* Italy bond yields fall after Thursday's sharp rise

* Euro zone PMIs stronger than expected

* Banks to repay just fraction of ultra-cheap ECB cash

* Euro zone periphery govt bond yields http://tmsnrt.rs/2ii2Bqr (Writes through)

LONDON, June 22 (Reuters) - Greek bond yields dropped on Friday after euro zone finance ministers agreed a plan to ease Greece's heavy debt burden and allow it to stand its own feet after it officially leaves a bailout programme in August.

Euro zone finance ministers extended maturities and deferred interest of a major part of their loans to Greece and agreed a big cash injection to ensure Athens is able manage a debt level that many feared was unsustainable.

"This had been anticipated but it is a positive in terms of the debt relief to Greece and the cash buffer that they will be afforded, which is important because it provides a safety net and allows them to follow the path of Portugal post-bailout," said Rabobank strategist Richard McGuire.

Greek 10-year bond yields fell 16 bps to 4.15 percent , its lowest level since mid-May, while five-year bond yields fell 19 bps to 3.29 percent.

But while the news undoubtedly bolstered Greek government bonds, it also seemed to contribute to buying of other Southern European debt, which has had a rough ride in recent weeks on concerns over Italian and German politics.

"It did clearly appear that Greek government bonds were leading the charge today," said Rabobank's McGuire. "It seems signs that EU policymakers are willing to come together to afford debt relief to Greece is a positive punctuation in what has been an otherwise gloomy period."

Italian government bonds in particular have suffered hefty losses in recent weeks, as an anti-establishment coalition government moved into power. Just the day before, Italian debt and the euro sold off on the appointment of two eurosceptics to head key finance committees in Italy's parliament.

But on Friday, Italy's 10-year bond yield fell 5 bps to 2.69 percent -- narrowing the gap over euro zone benchmark issuer Germany to 236 bps.

Bond yields across broader euro zone government debt markets faced some upward pressure from better-than-expected business activity data in the bloc.

IHS Markit's Euro Zone Composite Flash Purchasing Managers' Index, seen as a good guide to economic health, climbed in June to 54.8 from 54.1 in the previous month.

In Italy, Claudio Borghi, the president of the lower house budget committee, said in a newspaper interview on Friday that Italy's exit from the euro would solve many of the country's problems, but it is not part of the current government's plans.

Despite a calmer tone to markets, analysts said traders remained sensitive to the headlines coming from Italy - the euro zone's third-biggest economy.

René Albrecht, a rates strategist at DZ Bank, warned that sentiment is still fragile. "Although yesterday's sell-off was not as strong as the one in late May, traders are very sensitive to the noise," he said.

The European Central Bank meanwhile said on Friday that euro zone banks will repay 11 billion euros of ultra-cheap funding, returning only a fraction of their borrowings two years ahead of schedule.

Banks borrowed 399 billion euros ($465 billion) in a four-year targeted longer-term refinancing operation (TLTRO) in mid-2016, piling into a facility that promised to pay them a small amount of interest if they met their quotas for lending to the real economy.

(Reporting by Dhara Ranasinghe; Graphic by Balazs Koranyi; Editing by Catherine Evans)