By Jeffrey Heller

JERUSALEM, Oct 31 (Reuters) - Ehud Olmert is living the dream after a political nightmare of a war.

Less than three months after a costly conflict with Lebanese Hezbollah guerrillas turned him into one of its walking wounded, Israel's prime minister has rebounded to command one of the biggest parliamentary majorities in the country's history.

Along the way, the veteran politician has had to abandon past policy -- a signature plan for evacuating dozens of Jewish settlements in the occupied West Bank while strengthening others -- and shift public focus towards a strategic challenge, Iran.

With no peace prospects in sight with a Palestinian government led by the Islamist militant group Hamas, Olmert has won a key political ally, Yisrael Beitenu, a far-right party opposed to relinquishing occupied territory.

It joined what had been a fragile coalition on Monday, shoring up Olmert, whose popularity plummeted in opinion polls a after Hezbollah fired 4,000 rockets across the border in a 34-day war that ended without a decisive Israeli victory.

"A coalition of 78 legislators is the dream of every prime minister in this country," political commentator Yossi Verter wrote in the Haaretz daily, referring to the number of seats Olmert's government now controls in the 120-member parliament.

Olmert is now searching for an agenda after dumping the West Bank "realignment" proposal. The rockets fired from Lebanese territory Israel quit in 2000 raised fears among Israelis that Palestinians could launch similar attacks from evacuated land.

What Olmert does next with his expanded power baseis unclear.

PALESTINIAN PROBLEMS

On the Palestinian scene, Hamas and moderate President Mahmoud Abbas's Fatah are locked in a power struggle focused on the possible formation of a unity government both movements hope can ease a Western aid embargo.

Israel, backed by the United States, refuses to deal with Hamas unless the group recognises the Jewish state, accepts existing interim peace agreements and renounces violence.

And while Olmert has pledged to talk to Abbas, no meeting has been set. An Israeli soldier abducted by Gaza militants in June is still being held amid demands Israel release Palestinian prisoners, making a summit problematic for both leaders.

Instead, Olmert heads to Washington in about two weeks for his second White House session with U.S. President George W. Bush since winning election in March.

Addressing Congress during his previous visit, Olmert devoted a good portion of his speech to what he described as the "existential threat" Israel would face from a nuclear Iran.

A senior Israeli official said that is likely to be the main focus of his November 13 talks with Bush.

Israel has said it wants to take a back seat to international efforts to curb Iran's nuclear programme, which Tehran says is aimed at electricity generation.

But Olmert has said publicly that Israel has begun preparations for the day it fears Iran will build an atomic bomb and he has appointed Yisrael Beitenu's leader, Avigdor Lieberman, minister for strategic affairs.

Olmert also faces potential threats at home, including the eventual findings of a cabinet-appointed commission into the way his government and the military handled the Lebanon war. Israeli commentators say it could take a year before its report is in.

On another front, police have begun initial checks to decide whether to open a criminal investigation against Olmert over suspicions he promoted the interests of personal friends -- foreign-based millionaires -- in bidding for a state-owned bank.

He denies any wrongdoing in the privatisation of Bank Leumi last year or in two other affairs under examination by Israel's main government watchdog -- a private real estate deal and appointments of alleged Olmert cronies to a business authority.

Police investigations of alleged corruption by Israeli prime ministers have become the norm in recent years, dragging on for months and ending without criminal indictment.

In the meantime, Israelis are riveted to a sex crimes scandal involving President Moshe Katsav.

How rape allegations against the ceremonial leader play out will have little effect on Olmert's government but the media spotlight on Katsav could buy the prime minister more time to formulate his own strategy quietly for the future.

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