I. Irving Davidson is the consumate Washington lobbyist, with perfectly manicured nails, round gold cufflinks and "very dear friends" everywhere, from the Mafia to the Dallas Cowboys to the White House.

Billy Carter is, well, he's just Billy.

This is the story of how the two men happened to wind up together at last year's Super Bowl festivities in Miami.

This little slice of how things work in Washington comes, as might be expected, in conflicting versions. Davidson gave his this week to Senate investigators checking out a rumor that he might have played a role in putting Billy Carter in touch with the Libyan government.

Davidson denies the rumor unequivocally. Billy Carter first traveled to Libya in the fall of 1978. Davidson says he never met the president's brother until Billy and some friends turned up in Miami as Davidson's guests for Super Bowl XIII between the Pittsburgh Steelers and the Cowboys.

Carter and his friends showed up the night before the game for a cocktail party Davidson hosted aboard a Chinese junk tied up along a Miami waterway.

"I was asked if I could get him [Carter] into the Super Bowl," Davidson said in a telephone interview. He said "a very dear friend of mine . . . a lady who works at the White House," called him up a few days before the Jan. 21, 1979, football game and asked if he could get tickets for Billy and three friends.

Davidson, whose clients include Cowboys owner Clint Murchison, said he was glad to oblige -- and extended an invitation to the cocktail party as well. w

"I had about 70 people over, including some folks from Mexico and Peru," Davidson recalled. He said Billy, his sidekick, Henry R. (Randy) Coleman, and two other friends arrived around 8 or 9 p.m.

"The party ended around 10 p.m., but Carter and his friends didn't have a place to stay. There were several bedrooms on the junk so I let them have that." Davidson told a reporter. "The junk belongs to a friend of mine. It's still parked on the river down there. You want to use it sometime?"

Davidson said he had breakfast with Billy and his friends the next morning at the Palm Bay Club, where Davidson was staying, and that he lost track of them after that.

"I don't think they went to the game," he said. He suggested that Billy and his friends may have gone to a motel to watch the game. In any case, he said, Billy "was not in our group."

Davidson said he next saw Carter and his friends the day after the game, when he drove with them to Fort Lauderdale where Murchison was staying, to introduce them to the Texas millionaire.

"BSed on the way up," Davidson said. "They had a cup of coffee with Clint. But since that day, I've never seen [billy] or talked to him."

Davidson said the only business he has done with Libya concerned a ground-to-air communications system that one of Murchison's companies installed there years ago.

Davidson, who made headlines in June when he and reputed Mafia leader Carlos Marcello were indicted in New Orleans on fraud and racketeering charges, said he was asked in 1975 whether he could help supply uniforms for Libyan government workers. He said he recommended a New York firm, but "they didn't get the contract."

But as for reports that he might have helped get the Libyans and Billy Carter together, Davidson said: "Honest to God, I didn't."

The biggest difficulty with Davidson's account of events suggest how he has become such a successful "door-openere and arranger," as his business cards used to proclaim. His "very dear friend" at the White House, Mary Beazley, said she never called him and indicated that she does not know him that well.

Davidson "called me and said Mr. Murchison would like Billy Carter to attend the game as Mr. Murchison's guest," said Beazley, a former secretary to Jimmy Carter in Georgia and now the scheduling secretary for budget director James McIntyre. "I never even talked to Billy about it. I just told Randy Coleman [about the invitation], gave him Mr. Davidson's number and bowed out of the picture.

"I would never even have thought of asking for any invitation for Billy." Billy never asked me to do anything like that."

Davidson said he first met Beazley at a reception early in 1977. "We're very close friends to this very moment," he said.

Beazley said Davidson had her and her husband and "10 to 12 people I know" to a Redskins-Cowboys game a year ago. Asked whether she was "a very dear friend" of Davison, she paused and wondered: "How do you characterize somebody you just meet socially?"