THE HEADLINE in my recurring dream reads "Israeli Government Falls; Fifth in Five Months." The story outlines how Israeli Prime Minister Yitzhak Shamir (sometimes it's David Levy or Shimon Peres) has again resigned after a vote of no confidence in the Knesset. Shamir (Levy or Peres), in desperation, has called in a foreign adviser to tell him how to maintain order with the government of the month -- Amintore Fanfani, former prime minister of Italy.

It is, of course, only a fantasy, but unless Yitzhak Shamir has political talents so far kept well-hidden, it will not be long before 35 years of remarkable political stability in Israel comes to an end.

Not to put too fine an edge on it, Shamir has inherited a mess from Begin. What appeared a year ago from the Israeli government's perspective to be a glorious victory has turned into a bloody burden. Almost 200 Israelis -- roughly four a week -- have been killed in Lebanon since the heaviest fighting ended. Syria remains entrenched in the Bekaa Valley. Sectarian fighting among Lebanese factions has again broken out.

The Israeli economy is teetering. Exports are down, consumption is up. The foreign debt is rising and inflation is raging. The government has delayed taking effective measures to bring a potentially devastating situation under control.

Begin came to power by fashioning a coalition in the electorate of European and Oriental Jews, voters with little in common except their attraction to Begin and their hostility to the socialist Labor Party that governed Israel for its first 29 years. Without Begin, it is hard to see how this coalition can endure.

Begin was able to hold his coalition together, despite a political system that is inherently unstable, because he was a commanding figure within his Herut Party. In fact, Herut has been largely a one-man show since Israel became a state. Begin saw to it that potential opponents quit in frustration or were driven out. Former defense minister Ezer Weizman was only the latest in a long line of would- be successors to Begin to break with him.

Whatever else may be said about Begin, he was undeniably a man with appeal to the Israeli electorate. Shamir has no such hold on his party colleagues, and no following to speak of with the Israeli man and woman in the street.

The next Israeli government will have to take unpopular steps. But tough economic measures will threaten the coalition. The last time an effort was made to bring inflation under control, between 1979 and 1981, the austerity measures almost proved the undoing of Begin's government. But doing nothing will threaten Israel.

Shamir will be continually challenged by deputy prime minister David Levy, a 45-year-old Moroccan Jew with widespread popularity among North African Jews, the largest ethnic bloc of Israeli Jews. Levy, however, has yet to develop significant support among European Jews.

Begin's successor comes to power at a time when the consensus that bound Israel together has been severely weakened by the Begin government's settlements policy in the West Bank, which Shamir promises to continue, and by the invasion of Lebanon.

A generation ago, Israel's most illustrious political leader, David Ben- Gurion, stepped down as prime minister. He left behind a Labor Party rich in talented, if distinctly uncharismatic, leaders who saw a common interest in maintaining the status quo and containing individual ambitions. Labor had an ideology and its own powerful institutions: unions, business offshoots and a health plan in which some 85 percent of all Israelis were enrolled. Indeed, one of the reasons Labor ultimately was voted out of office was that it had come to confuse the party's welfare with the country's.

Herut lacks the kind of institutional vested interest that Labor had then, and still has today. By systematically eliminating potential rivals, Begin precluded the possibility that a popular successor could be groomed.

On top of all of this is Israel's own form of parliamentary democracy, which allows any party with 1 percent or more of the vote to be represented in the Knesset. This rather pure form of proportional representation encourages the development of splinter parties and virtually ensures a situation in which no single party can win enough votes to form a government without a coalition. No party has ever won a majority of the vote in an election. So small parties can always exact a price from the dominant party for ruling.

The Labor Party has problems of its own, not the least of which are divisions over what Israel's policy should be on the West Bank and a vicious, often public battle between party leader Shimon Peres and his predecessor in that position, former prime minister Yitzhak Rabin.

But Labor's heaviest burden is its own refusal to act like an opposition party. Though out of power for six years now, Labor still acts like the government in exile. In the public mind, for better or worse, Labor continues to be identified as the Establishment. Labor now generates little enthusiasm among the professional middle-class, which once supported it but abandoned it in 1977 because the party had grown corrupt, arrogant, inflexible and unimaginative. And Labor continues to arouse hostility among Oriental Jews, especially those from North Africa, who remember being treated as second- class citizens by a succession of Labor governments.

It seems likely that the smaller parties will swing back and forth between Herut and Labor, providing first one and then the other with a majority, only to defect from the coalition when the government fails to perform. Or, elements of the ruling party will defect when the government honors aspects of the coalition agreement that they find obnoxious.

In the old days, Menachem Begin would rally the troops in such situations. But Begin is gone and we are not likely to see an Israeli leader with his authority for some time to come.