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Whatever riddles had surrounded the deaths in Arizona last year of former area resident Julie Sweeney and her lover, police said they believe most have been solved: They say they know their suspect, his motives, how he killed the couple and when.

Police have wasted little time looking in other directions since the bodies of Sweeney, 32, a married mother of two, and Allan Neuman were found shot to death in Neuman’s Scottsdale home on Nov. 17, 1998.

But even now, after the anniversary of the murders has come and gone, investigators are stuck. Short of a confession, they say, closing the case may hinge entirely on tracing the 30-year-old life of a Rohm R.G.31, a rare five-shot revolver found at the scene, and somehow placing it in the hands of their suspect.

“Ownership of that gun is crucial,” said Sgt. Doug Dirren, spokesman for the Scottsdale police. “It will help corroborate the crime scene evidence and establish a solid link to the suspect.”

So far, investigators have had little luck, tracing the gun no farther than a Virginia gun shop that closed years ago and dumped all inventory records.

To fill in the blanks, investigators have been showing photos of the gun and making contacts with police and merchants in cities, including Chicago, where they believe the suspect traveled in the weeks before the killings.

Julie Sweeney and her husband, Tom, had moved to Arizona from Schaumburg in mid-1996, settling in the Phoenix area with their two young children. Not long after, their marriage appeared to be collapsing. And by spring 1998, Julie Sweeney’s relationship with Neuman, 48, a successful colleague of hers at a nutritional-supply company, had gone from friendship to something more.

That September, she filed for divorce. But in statements to police and friends, Tom Sweeney, a former real estate budget analyst, said that the marriage was on the mend in November and that Julie Sweeney had made that last trip to Neuman’s home to cut off the affair.

When his wife failed to return, Tom Sweeney told police, he hopped the fence outside Neuman’s north Scottsdale home and entered a back door.

On a living room couch, he told police, Sweeney found his wife curled up next to Neuman, who was sitting upright with his head cocked back and his right hand wrapped around the gun. Both had been dead as long as 24 hours.

“I’ve got like a murder . . . two dead bodies . . . a murder-suicide thing,” he said in a 911 call.

At the outset, authorities gave little hint that they disputed Sweeney’s story.

Autopsies were done and the medical examiner ruled that the evidence suggested Neuman had killed his lover, then turned the gun on himself. Family members of both victims quickly denounced the findings, saying it was unfathomable that gregarious, gentle Neuman would commit such a desperate act.

Six weeks later, the case began to turn. The medical examiner changed the ruling to a double homicide.

Warrants were served on Tom Sweeney’s home, office and car, and while police have not named Sweeney as the killer, they have broadly hinted that he is their chief suspect.

Sweeney, 39, could not be reached for comment. Lawrence Kazan, a noted Phoenix-area criminal attorney representing Sweeney, on Friday denied any involvement by his client.

“There’s no evidence other than their speculation,” Kazan said. “Our system is much more substantive than someone being fingered for a crime because of speculation.”

Though Kazan said he was unaware of any evidence tying his client to the crime, he said police have been digging for a link.

Neuman’s brother Henry recently returned to New York from a trip to Arizona, where the two families met with investigators and gathered for memorials.

“This has become a major priority in my life, and I will not stop,” Henry Neuman said, adding that “anger and frustration, disappointment” linger with both families.

“The most incredible thing is they know exactly who committed the crime. Not only do they know who, they know how. This is an open wound.”