New fighting erupted today as Syrian troops rejected an Israeli ultimatum to pull out of the Lebanese capital and moved to protect their positions against Israeli assaults. Clashes were reported between the two armies in the hills overlooking Beirut's now closed airport, and other battles broke out in Palestinian neighborhoods in Beirut.

The scale of the fighting was not immediately clear nor were casualties announced. A cease-fire, still formally in effect, was declared between the Syrian forces Friday and widened on Saturday to include the Palestine Liberation Organization.

Lebanese state television said Syrian and Israeli tanks battled for more than four hours near the airport. Lebanese radio stations said the Syrians again attacked when an Israeli armored column moved through the Syrian-held town of Shweifat, a mile east of the airport.

Before the tank fight, Israel issued a demand for the remaining Syrian troops to leave the capital and its environs. It was couched in the form of an ultimatum--with no specified time limit--and conveyed to the Syrians through Lebanese Brigadier Sami Khatib, the nominal commander of the now battered all-Syrian Arab Deterrent Force. Syria rejected the demand, noting its 1976 Arab League mandate to act as a peace-keeping force in Lebanon. See related story on page A21.

On the political front, U.S. special envoy Philip C. Habib conducted intensive negotiations here with Lebanese Prime Minister Shafiq Wazzan, Foreign Minister Fuad Butros and several Lebanese politicians.

As usual, Habib said nothing about the talks. Carrying forward his shuttle diplomacy begun in Syria last week, Habib was reliably reported to be concerned with discussing Israel's conditions for a withdrawal of the 25,000 troops that invaded Lebanon June 6 and averting renewed fighting between Israeli forces and Palestinian guerrillas, who are pinned down together with about a half-million Lebanese in predominately Moslem West Beirut.

The first visible result of the Habib mission was the departure from the hill suburb of Baabda, site of the presidential palace, of all but a handful of the Israeli armored vehicles stationed there since Sunday when they closed the ring on West Beirut.

Habib was reliably reported to have been angered by the Israeli occupation of Baabda, a move that many Lebanese described as an intense humiliation.

No progress was visible in President Elias Sarkis' continuing efforts to reconcile squabbling Lebanese politicians and warlords to put together a national salvation council to run the country.

The prime objective of such a council would have involved moving units of the newly revived, 21,000-man Lebanese Army into West Beirut to reassert national authority and to dissuade the Israelis from rooting out their Palestinian enemies.

Druze leader Walid Jumblatt continued to thwart the Sarkis plan by refusing to join the council, on which he was to have represented both his Druze community and the left-wing National Movement militias.

Saeb Salam, many times prime minister, said in an interview that the council should start acting in Jumblatt's absence and indicated that Yasser Arafat, the PLO chairman, had backed the project.

There was a mood of growing despair among politicians and ordinary citizens who feared that if the council did not quickly begin to act, the Israelis would resume full-scale hostilities despite their formal pledges to spare West Beirut.

PLO spokesman Mahmoud Labadi today repeated earlier PLO assertions that the guerrillas were in good condition and unbeaten. He also denied Israeli reports that Arafat had sought refuge in a foreign embassy.

Other unconfirmed reports here said that Arafat had been spending nights in the Soviet Embassy, leaving during the day to provide visible leadership to his forces on the streets of Beirut.

In fighting today, Lebanese security forces reported that two Lebanese soldiers were killed when their jeep was attacked early this morning at Kfarchima, near the airport south of the city. The assailants were variously described as Syrians and Palestinians.

In midafternoon, Lebanese sources reported a brief, minor clash between Syrians and Christian militiamen in the same general area when the Syrian troops tried to advance.

It was the first such clash involving the Lebanese militia since the invasion began 10 days ago. Later, four Syrian tanks were knocked out by Israeli gunfire in the same area, the sources said.

Early this morning, Syrian and Israeli troops cut the key Beirut-Damascus highway and were reported by Lebanese Army sources to have set up positions less than a mile from each other.

There was no word of the progress of the Israeli armored column that yesterday crossed through territory of the Christian Maronite militia, allied with the Israelis. That column was reported by Lebanese officials to be moving eastward to drive the Syrians and their leftist Lebanese allies from the mountain peaks overlooking the Mediterranean and the Bekaa Valley to the east.

The Syrians captured those peaks in bitter fighting last year with the Christian militia. The fighting led to a major crisis involving the installation by Syria of ground-to-air missiles in the Bekaa Valley.

The Israeli drive through Christian territory apparently caused some second thoughts among Christians who long had favored an Israeli invasion to deliver them from their common foes, the Syrians and Palestinians.

"The Christians are beginning to understand," one diplomat remarked, "that they cannot use the Israelis and that the Israelis use other people."

According to unconfirmed reports in Christian East Beirut, Lebanese forces commander Bashir Gemayel first learned of Israeli intentions to enter his territory when Israeli Defense Minister Ariel Sharon conferred with him early Monday morning. According to some sources who claim inside knowledge, Gemayel was said not to have been amused.