(FORTUNE Magazine) – H. Ross Perot Jr., namesake of the Texas billionaire and CEO of Perot Systems, a FORTUNE 1,000 data-services company, has always been an aviation nut. Back in 1982, when Ross Jr. (as he's known around Dallas) was a mere 23, he claimed one of the more pointless world records, spending millions in family money to co-pilot the first helicopter flown around the world. In his 30s he built a successful industrial airport in Fort Worth. Now, at 45, Perot's passion for flying has gotten him into trouble--generating a federal investigation and a dramatic split with his company's longtime law firm.

The problem dates back about two years, when Perot, operating through the Alliance Heritage Museum, a nonprofit entity he established, got his hands on a pair of dilapidated Air Force T-38 Talon supersonic trainer jets. Perot--a former Air Force pilot--reportedly spent about $5 million to restore one of the T-38s with the aim of flying it himself. But under federal law, planes formerly flown by the U.S. military are supposed to remain grounded and in the secure possession of an officially sanctioned military museum or civic group.

Neither Perot nor his planned "museum"--which existed only on paper--met either requirement, prompting the military to refer the matter to the U.S. Attorney's office in Dayton, where prosecutors have been quietly investigating for more than a year. The feds demanded that Perot turn over the rebuilt jet--either to the government or to an approved museum. "The Air Force believes the government owns this plane," said Fred Alverson, a spokesman for the Ohio U.S. Attorney's office, a few months after the investigation began. "The people at [Perot's Alliance Heritage] museum think they own it. Whether that difference will lead to civil or criminal proceedings is somewhere in the second act, which hasn't played yet."

Perot, meanwhile (who declined to comment), hired a criminal lawyer. Perot Systems retained special outside counsel, Covington & Burling, to investigate the "facts surrounding the T-38 matter," according to company spokesman Eddie Reeves. Covington & Burling soon concluded that the publicly held company wasn't required to reveal the matter in its SEC filings. But Perot Systems' primary outside law firm, Hughes & Luce, disagreed, insisting that the continuing federal investigation required public disclosure. When Perot Systems refused to follow that advice, the respected Dallas law firm--which has ties to the Perot family dating back more than 20 years--took the extraordinary step of severing its relationship with the company.

Earlier this year Perot finally delivered one T-38 to the Frontiers of Flight Museum at Dallas's Love Field airport and agreed to ship the second T-38 to a government plane graveyard. All this has seemed to set the stage for a quiet settlement, though the agreement has yet to be finalized.

What does the Perot Systems board think about the behavior of its CEO? Board chairman H. Ross Perot Sr., through his secretary, declined comment.

--Miriam Rozen