Talks to reunify Cyprus are to resume after the resolution of an impasse at a meeting of the island’s Greek and Turkish leaders.

Emerging from a four-hour meeting with Nicos Anastasiades and Mustafa Akıncı in New York on Sunday night, the UN secretary general, António Guterres, said negotiations would recommence later this month. The breakthrough was met with relief on the island, where many had feared the collapse of talks widely seen as a once-in-a generation chance of reconciliation.

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Flanked by island’s two leaders, Guterres said all three had “agreed on the need to reconvene the conference on Cyprus”. A special UN adviser, Espen Barth Eide, who terminated mediation efforts last month in exasperation after two years of shuttle diplomacy, would be tasked with the “preparation of common documents to guide discussions on security”, he said.

“The chapter on security and guarantees is of vital importance,” Guterres continued, emphasising that both were essential to building trust between the two communities in an envisaged federation. “The leaders agreed [that] … all other outstanding issues, starting with territory, property and governance and power sharing … will be negotiated interdependently and that nothing is agreed till everything is agreed.”

Cyprus has been divided since the summer of 1974 when Ankara, responding to an attempted coup to unite the island with Greece, ordered an invasion, with Turkish troops seizing its northern third. No other country but Turkey recognises the self-proclaimed breakaway republic in the north.

The former US vice-president Joe Biden, who took a personal interest in achieving a settlement during the Obama administration, insisted a solution was “within reach” at the weekend, saying it would be of considerable benefit to all Cypriots.

“We have a unique window of opportunity now to resolve the conflict in Cyprus so that future generations of Cypriots can live in peace, harmony and prosperity,” he told Greek newspaper Kathimerini before a visit to Athens this week.

“The two sides are close to resolution … A settlement will not only open up financial resources from the international community and private sector, it could also help maximise the economic benefits of Cyprus’ offshore hydrocarbon reserves.”

Guterres did not say when the talks would resume. An exact date, he said, would depend on consultations with the three powers guaranteeing the island’s security – Greece, Turkey and the UK. A first conference in Geneva in January had been attended by foreign ministers from all three countries. Diplomats had said that when talks reconvened in the Swiss city, they would be at prime ministerial level, because by then a solution would be judged “to be at hand”.

A prompt return to negotiations is considered essential because campaigning will soon begin for the 2018 presidential election in the Republic of Cyprus, which Anastasiades is expected to contest.

Speaking from Nicosia, the island’s partitioned capital, the political commentator Kyriakos Pierides told the Guardian: “Thanks to the heavy engagement of the United Nations secretary general, the deadlock has been broken.

“It is very welcome. There had been a lot of disappointment not only in the collapse of talks, but collapse of confidence between the two leaders. To get to a solution we don’t just need to give and take. It’s a matter of pure political will, of courage and leadership that both must show.”

Security is by far the biggest impediment to a solution. Greek Cypriots want the removal of an estimated 40,000 mainland Turkish troops from the north of the island and the abolition of a security system that allows a guarantor power to intervene if the safety of a community is deemed to be endangered.

Anastasiades has proposed an international police force to oversee post-reunification security, claiming Cyprus’s EU membership provides adequate security to prevent inter-ethnic strife from erupting again.

Hubert Faustmann, a professor of history and political science at the University of Nicosia, said: “The Turkish Cypriots have agreed to modification of the system, but not its abolition. The demands of both are mutually exclusive, but now the two sides have been cornered into a situation where a final and decisive decision has to be taken and nothing can be ruled out.”