(Adds Olmert comments)

By Jeffrey Heller

JERUSALEM, July 3 (Reuters) - A deadly rampage in a bulldozer by a Palestinian resident of Jerusalem left Israel grappling on Thursday with the dilemma of how to maintain security in the city along with the premise it is undivided.

Israel captured East Jerusalem in the 1967 Middle East war and annexed it along with nearby villages in a move that is not recognised internationally, granting Palestinian residents Israeli identity cards that gave them wide freedom of movement.

In issuing the same documents used by Jews, Israel was sending a signal that East Jerusalem -- which Palestinians want as the capital of a future state in the West Bank and Gaza Strip -- was part of the "indivisible capital" of the Jewish state.

But Wednesday's attack on Jerusalem's busy Jaffa Road in which three Israelis died and a shooting spree, also in Jewish west Jerusalem, which killed eight Israelis in a religious seminary in March have combined to raise particular concern.

Both attacks were carried out by Palestinians from areas Israel regards as part of East Jerusalem. Unlike Palestinians from the West Bank, where Israel has built a controversial barrier, the two men could work and travel in all of Jerusalem.

Prime Minister Ehud Olmert proposed after Wednesday's incident the demolition of the homes of Arab East Jerusalem residents who carry out attacks against Israelis.

"I think we need to be tougher in some of the means we use against perpetrators of terror," Olmert told an economic conference in the southern port city of Eilat.

"If we have to destroy houses, then we must do so, and if we have to stop their social benefits, then we must do so. There cannot be a case where they massacre us and at the same time they get all the privileges that our society provides," he said.

But demolition notices would likely draw legal countermoves by Palestinians from East Jerusalem in Israeli courts, as well as international protests that destroying a home the attacker shared with other family members was collective punishment.

"Demolishing the home is not the answer. This is punishment for the family, which had nothing to do with this," Imad Muna, a 44-year-old resident of East Jerusalem, said in Hebrew, noting the bulldozer driver was shot dead during the attack.

BARRIER

Israeli Vice Premier Haim Ramon said re-routing the West Bank barrier to include Palestinian villages Israel considers to be part of East Jerusalem could be a solution. The two attackers came from such communities.

"These are Palestinian villages that were never part of Jerusalem. They were annexed to it in 1967," Ramon said on Army Radio, echoing comments Olmert made several months ago.

However, redrawing the barrier's route would draw fire from Israel's right wing and entail a change in government policy at a time when borders are a central issue of peace talks.

Tightening security in and around East Jerusalem could also be difficult and give the impression of a physical divide in the holy city -- something Israel has long sought to avoid.

"There are 200,000 Arabs in East Jerusalem. You can't put up roadblocks or a fence that would make life unbearable for everyone," one Israeli government official said.

Many of Jerusalem's Jews and Arabs live in neighbourhoods that abut each other. They shop at the same malls and share government services.

"(The attack) will hurt co-existence in the city," said Shira Ezra, an 18-year-old Israeli student.

But Muna predicted no radical change in relations between the two communities.

"Once in the while, a crazy person does this kind of thing. But there is co-existence. Most of the people are already at peace," the East Jerusalem resident said. (Additional reporting by Avida Landau, Editing by Diana Abdallah)

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