The conflict has claimed more than 500 lives since September

Israeli leaders have indicated they are prepared to maintain a unilateral ceasefire ordered last week, despite two car bomb attacks in Jerusalem carried out by Palestinian militant groups on Sunday.

Mr Burns' task has not been eased by these attacks, which injured no one seriously, or by further incursions by Israeli tanks in the Gaza Strip on Monday.

Violence overshadowed a first round of talks on Sunday, when the US envoy discussed the recommendations of the Mitchell inquiry into the current violence, which calls for an immediate and unconditional ceasefire.

Mr Burns urged Palestinian leader Yasser Arafat to do all he could to stop militants attacking Israeli targets.

We had a constructive meeting about the implementation of the Mitchell report on ending the violence

William Burns

Senior figures on both sides said the mere fact that the US had appointed Mr Burns was a positive sign that the Bush administration intended to involve itself in Middle East diplomacy.

But the BBC's Simon Ingram in Jerusalem says that at this stage both Israelis and Palestinians are essentially telling the Americans what they want to hear, while refusing to make the sort of concessions that might bridge the gaps between them.

Incursions

Monday began with fresh incidents in Gaza. Israeli tanks and bulldozers entered Palestinian territory east of Karni and levelled olive groves. The army said it was conducting repairs to a border fence.

The blasts fuel fears that the violence may again spin out of control

Last week, Israeli Prime Minister Ariel Sharon banned troops from carrying out raids into Palestinian autonomous areas or from firing upon Palestinians unless Israeli lives were at risk.

The Palestinians accuse Israel of 96 violations since his announcement.

Sunday's two car bomb explosions in Jerusalem were the latest in a recent spate of such attacks.

They occurred in a shopping and entertainment area close to police headquarters, just hours before Mr Burns met Mr Arafat.

Police said nobody was injured in the first, while two were slightly hurt in the second explosion about nine hours later.

The Popular Front for the Liberation of Palestine said it carried out the first attack, while the Islamic Jihad movement said it was responsible for the second.

Mr Arafat has said he can only rein in the militants and Palestinian anger when Israel ends the building of Jewish settlements.