In the early morning hours of May 23, unidentified intruders broke into the Port-au-Prince home of Dr. Paul J. Alexander, an American physician who was just completing a rural health project funded by the U.S. Agency for International Development (AID) in Haiti.

The assailants put the family dog outside, cut the cord of a ground-floor telephone, then entered the second-floor bedroom where Alexander lay asleep next to his wife, Jean. As flashlights were pointed into the couple's eyes, one of the men fired a 22-cal. bullet directly into Alexander's heart. Then they fled as mysteriously as they had entered, shouting to each other in Haitian Creole.

Haitian officials investigating Alexander's murder initially described it as a foiled burglary attempt, or a spinoff of the often-wanton violence that accompanied the overthrow of Jean-Claude Duvalier's second-generation dictatorship.

But family members and business associates, who have expressed frustration at the slow pace of the police investigation, fear another, more ominous possibility: that Alexander was the target of an assassination, connected somehow with his work of reorganizing Haiti's national health service.

"The project itself would have been very unpopular with anyone whose economic interests would have been threatened," said Peter Rousselle, vice president of the Boston-based Management Science for Health (MSH), Alexander's employer.

The five-year project, conducted by MSH under contract from AID, was aimed at strengthening the management system of the Haitian ministry of health. It involved setting up a system of community pharmacies in poor, rural areas that would purchase drugs and sell them at discount prices.

Apparently now even the Haitian police have dropped the burglary explanation. Sources in Port-au-Prince who are familiar with the ongoing probe said police now believe that man who shot Alexander went to Haiti from the United States, probably Miami, and returned shortly after the murder. "The police are not using the word 'burglary,' " one informed source reached by telephone from Washington said.

At least three theories, all based on circumstantial evidence and speculation, have emerged. Was Alexander murdered by powerful elements in Haiti's largest pharmaceutical industry, which lost a bid for the $200,000 contract to supply the new rural clinics? Was the murder the work of employes who stood to lose their jobs once Alexander's project was completed? Or had Alexander uncovered some misuse of funds -- possibly U.S. aid -- in the health ministry?

"I know there's something here, it's just too bizarre," said David Alexander, his son and a physician.

"It is extremely frustrating," he said. "Nobody is really doing anything. A boy is killed in Chile, and the State Department is all over the Chileans to do something. My father is killed in Haiti, and they keep telling me that it's in the hands of the Haitian police."

David Alexander and the other family members have written repeatedly to officials in the State Department, and in the U.S. Embassy in Port-au-Prince, asking them to put pressure on Haitian authorities to intensify the probe. They have also enlisted the aid of Rep. James L. Oberstar (D-Minn.), who represents Alexander's home in Minnesota and who met Paul Alexander during a congressional trip to Haiti in April.

U.S. officials, in reply, said the matter is an internal Haitian police investigation, for which the United States can only give technical assistance if asked. In Washington, a State Department spokeswoman said, "They have to do it; we can't go in there and do it."

Said one source in Haiti who is familiar with the police probe: "It's a Third World police force and they're overwhelmed. They don't have very sophisticated methods."

One controversy that Paul Alexander before his death was involved in centered on his work with AGAPCO, a semiautonomous entity set up to buy drugs and distribute them at low cost to the rural population. In December last year, AGAPCO called for bids for a quarter-million-dollar contract for drug shipments. Haiti's leading pharmaceutical house, PHARVAL, offered a bid but was undercut by two new companies headed by the son of Duvalier's health minister.

In letters to his son, Paul Alexander described "the major tempest in our teapot" caused by the award of the contract, and how PHARVAL accused AGAPCO of collusion and impropriety. According to a letter Paul Alexander wrote Jan. 25, a PHARVAL official accused AGAPCO of "overstepping its official mission, destroying local industry, and contributing to the unemployment in the country."