By Ori Lewis

JERUSALEM, Jan 25 (Reuters) - Two Palestinians and an Israeli border policeman were shot dead in violence in the occupied West Bank on Thursday.

In Gaza, more people streamed across the breached border into Egypt to escape the strictures of an Israeli blockade.

Israel's deputy defence minister said the Jewish state wanted to cut its links with the Hamas-controlled Gaza Strip after militants blasted open the territory's border fence with Egypt.

U.S. Secretary of State Condoleezza Rice said it was up to Egypt to ensure the security of its border with Gaza.

"It's a difficult situation for them, but it is an international border. It needs to be protected and I believe that Egypt understands the importance of doing that," she said.

Washington said it was willing to work with Egyptian authorities to restore order but did not give details.

In the early hours of Friday, an Israeli air strike on a car killed at least two Palestinian militants in Rafah, Gaza security officials and medical staff said.

Jewish settlers shot dead two Palestinians and gunmen killed an Israeli border policeman in two separate incidents in the occupied West Bank on Thursday night in areas which have remained relatively calm for a months.

A police spokesman said Palestinian gunmen shot an Israeli paramilitary border policeman and seriously wounded a female colleague at the Shuafat refugee camp in the West Bank near Jerusalem.

The Israeli was the first fatality in the occupied West Bank since Palestinian gunmen shot and killed two off-duty soldiers near the city of Hebron last month.

There was no immediate claim for the attack.

In a second incident Jewish settlers overpowered and shot dead two Palestinians who infiltrated the settlement of Kfar Etzion not far from Bethlehem, an army spokeswoman said.

Deputy Defence Minister Matan Vilnai said on Thursday Israel wanted to wash its hands of Gaza altogether by handing over the supply of electricity, water and medicine to others. An Israeli security official said Egypt should take over responsibility.

"When Gaza is open to the other side we lose responsibility for it. So we want to disconnect from it," Vilnai said.

Israel occupied Gaza in 1967 but pulled troops and settlers out in 2005, although it still controls the strip's northern and eastern borders, airspace and coastal waters. It has imposed a blockade it says is meant to counter militant rocket fire.

A spokesman for Hamas, which seized control of Gaza after routing Palestinian President Mahmoud Abbas's Fatah forces in June, rejected Vilnai's disengagement idea as an attempt to separate Gaza from the occupied West Bank.

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