For officials in the State Department, the Billy Carter controversy is more than a messy public incident: It is an incident that could, they fear, worsen America's already touchy relations with Libya, the third-supplier of crude oil to the United States.

U.S. ties with Libya have been volatile. In the last eight months alone, for example, Libyans have stormed and burned the American embassy in Tripoli, expelled dozens of Americans and vigorously denounced the American-sponsored Mideast peace effort. The United States, in turn, in May booted four Libyan diplomats out of this country, saying they were involved in a campaign to kill exiled opponents of Libyan leader Col. Muammar Qaddafi.

Indeed, says one State Department official, "I don't see how our relations could get any worse."

But others express concern that the Billy Carter controversy could end a brief period of relative quiet that has prevailed since May, when Libya, after protesting for days, finally recalled the four diplomats the United States had ordered out of the country.

"When there is a churning of the waters, it can cause its own set of problems," said one official who asked not to be quoted by name."What we were trying to achieve was just keeping things quiet for a while."

During Qaddafi's 11-year revolutionary rule, America's relationship with Libya has operated on two very different levels.

On one hand, the United States and the North African state of 2.7 million people have an extensive and mutually beneficial economic relationship.

American gets 8 percent of its imported oil from Libya, 650,000 barrels a day. Only Saudi Arabia and Nigeria export more to the United States. And it is light, low-sulfur oil, easily and profitably refined into gasoline and not available from many other nations. U.S. oil companies operating in Libya -- including Mobil, Exxon, Occidental, and Conoco -- make more from their Libyan operations than those in almost any other member of the Organization of Petroleum Exporting Countries.

For its part, Libya in 1979 grossed almost $5.5 billion from its exports to the United States, and rising prices are expected to push that figure to $9 billion this year. In addition, Quaddafi's nation depends heavily on U.S. oil technology. Last year alone, the Libyans purchased $330 million in heavy equipment.

On the other hand, a string of incidents and policy disagreements between the two nations has left diplomatic relations so rocky that a State Department official now sees progress in that "nothing really bad has happened lately."

Qaddafi's government has been in constant conflict with U.S. notions of how nations should behave.

Libya has helped arm and finance international terrorist groups; Qaddafi has armed his nation to the teeth -- largely with Soviet weapons -- and sought nuclear capability; his troops fought unsuccessfully to save the regime of Ugandan dictator Idi Amin, who later took refuge in Libya; and Qaddafi has repeatedly denounced U.S. Mideast policy, especially the Egyptian-Israeli peace treaty.

Then there have been the incidents. In 1972, Qaddafi exercised an option to buy eight American C130 cargo planes for $4.5 million apiece, only to have the United States then refuse to grant export licenses for them. The United States said C130s Libya purchased in 1969 had been used for terrorist activities. In January 1979, Libya tried again to buy American planes -- this time three Boeing 747s. But the U.S. government blocked that $186 million purchase as well, saying Libya had recently used similar passenger planes to aid terrorists.

In December 1979, a mob of 2,000 Libyans stormed and burned the U.S. Embassy in Tripoli. The United States pulled all diplomatic personnel out of the country and threatened further action until Libya agreed to pay for the damage. The embassy remains empty.

In May of this year the United States ordered four Libyan diplomats to leave the country, alleging that they were involved in a Qaddafi-inspired campaign to kill Libyan dissidents here who refused to return home, a charge that had resulted in the expulsion of two Libyans the previous month. The four diplomats defied the order for more than a week, holing up in the Libyan mission here. Qaddafi threatened to cut oil exports to the United States, before agreeing to recall the diplomats.

A few days later, Qaddafi expelled 25 of the 2,500 Americans in Libya. A broadcast by the official Voice of the Arab Homeland said the Americans, mostly oil company employes, were deported for "spying and having connections with terrorist organizations."

Given the volatility of U.S.-Libya relations, it is not surprising to find some State Department officials concerned that the Billy Carter affair could upset the relative peace between the two countries.

Now "a manageable little problem," it will only cause problems if it drags on, said one State Department official. But like the entire relationship, as he noted, "it's very uncertain."