Greece extended a deadline for a second time on Thursday for remaining bondholders to accept a debt swap, giving Athens a little more breathing space to formulate a response to investors who have refused to sign up for the landmark deal.

Athens must in any case make a decision soon, in coordination with its EU and IMF backers, because a 450 million euro bond expires on May 15, finance ministry and banking officials said.

Completion of the bulk of the swap allowed Greece to get a new EU/IMF bailout last month and has helped ease market concerns over the euro zone debt crisis, at least for now.

But how Greece deals with the remaining investors is likely to set an important precedent for other highly indebted euro zone states and vulture funds who buy distressed government debt in the hope of securing a bigger payout in courts.

Athens gave investors on Thursday until April 20 to join in, extending a deadline that had already been moved back once to April 4.

A majority of investors have already rejected the debt swap for the May 15 bond, a finance ministry official said. But they cannot derail the overall debt swap plan - the biggest debt restructuring in history - which has now been accepted by about 97 percent of investors.

A source familiar with the talks said that the majority of the bonds maturing in May are in the hands of activist shareholder Elliott Advisors, which is well-known for contesting previous sovereign debt restructurings in Latin America. Elliott declined to comment.

Greece has said it cannot afford to fully pay holdouts and that the swap deal that domestic-law bound bondholders were forced to accept last month is the best available offer.

Outside the Bank of Greece, the country's central bank, about 50 retail bondholders protested on Thursday for not being excluded from the writedown.

ATHENS' THREE OPTIONS

The government is left with three options to confront those bondholders still resisting: continue to service the bonds, default and trigger litigation, or come up with a new offer while ensuring fair treatment for those that accepted the swap.

May 15 is the latest deadline for Greece to decide what it will do with the bondholders who don't want to participate in the PSI, a treasurer at a big Greek bank said on Thursday, referring to the debt swap, also known a PSI.

Greece completed the bulk of its bond exchange on March 12, swapping a nominal amount of 177 billion euros ($235 billion) of domestic-law government paper for new securities, inflicting real losses of about 74 percent on private-sector bondholders.

Athens said it would settle on April 11 the exchange of 20.27 billion euros of debt, equivalent to 72 percent of the bonds issued under foreign law and state enterprise notes signed up for the swap offer.

Bondholder meetings were held on March 27-29 for holders of 36 bond issues to vote on the debt swap. Meetings regarding seven of the bonds, representing a face value of 2.45 billion euros, were adjourned with no vote. Their April 18 deadline was also extended on Thursday, to April 20.

Two other government-guaranteed bonds, worth about 450 million euros in total and issued by Greece's state railway company under foreign law, expire in the second half of the year, a finance ministry official said.

(Additional reporting by Sophie Sassard and Sarah White in London; Writing by Ingrid Melander and George Georgiopoulos; Editing by Susan Fenton)