Greece and Macedonia failed on Friday to solve their dispute over the latter's name, but vowed to continue talks on the row which threatens Macedonia's chances of being invited to join NATO in April, according to dpa. "Both sides declared themselves ready to hold further talks under the auspices of the UN in New York in the coming week," a brief statement from the Greek side ran, according to NET Greek state TV. Greek Foreign Minister Dora Bakoyannis and her Macedonian counterpart, Antonio Milososki, met in Brussels with US Under-Secretary of State Daniel Fried to discuss the issue of what Macedonia - internationally known as the Former Yugoslav Republic of Macedonia or FYROM - should be allowed to call itself. Macedonians insist that their country should be referred to simply as Macedonia, the name by which it is known in their constitution. But Athens says that the geographical territory known as "Macedonia" stretches into both Greece and Bulgaria - and that letting one country use the name would allow it to make territorial claims on its neighbours. The name issue has dogged relations between Athens and Skopje since Macedonia broke away from Yugoslavia in 1991. However, it has now become urgent, because NATO leaders want to invite Macedonia to join the alliance at a summit in Bucharest on April 2-4. Greece, as a current member of NATO, says that while it supports Macedonia's candidacy, it will have no choice but to veto the invitation unless the name issue is solved to its satisfaction. UN mediator Matthew Nimetz recently proposed a number of alternative names, which reportedly included the New Republic of Macedonia, Upper Macedonia and the Republic of Macedonia (Skopje), but no decision has yet been reached on that point. NATO members agree that the name issue is a bilateral problem, but many of the bloc's 26 states say they are concerned by the dispute. US State Department spokesman Sean McCormack on Thursday in Washington said that the US "would certainly like to see it solved." "It's time to find a resolution to the issue. And there's no reason that these two states, with reasonable, elected democratic leadership, can't find a reasonable solution to the issue. We understand it's a emotional. We understand it's hard. But they can find a solution," he said.