June 10 was a busy, important day for President Carter's brother Billy, though it isn't known yet just why. On that Tuesday, a lawyer representing Billy Carter telephoned the Department of Justice to say that Carter would be in Washington the next day, and would like to check into the status of his case at the department. An appointment was quickly made for June 11.

Also on June 10, Billy Carter called his brother's national security adviser, Zbigniew Brzezinski, to say he'd like to come to see him. Brzezinski made an appointment to see Carter the next day, June 11.

Why did Billy Carter and his lawyer make those calls that day? The only answer on the record so far is vague. The full answer might explain what the latest flap over the president's brother really adds up to

The inference that Billy Carter had been alerted by June 10 that he had a serious new problem is all but irresistible. Until then, he had shown no inclination to cooperate with the Justice Department's efforts to investigate his relationship with Libya. In early 1979 he simply ignored two letters from the department asking about his ties to Libya. In january of this year federal investigators came to talk to him, and Carter denied that he was a paid agent to Libya. (Sometime during the same month of January, Carter received his first cash payment from Libya, $20,000.)

Carter's attitude changed just eight days after the Justice Department learned that Billy had received payments totaling $220,000 from the Libyan government. That happened on June 2, when the department learned from what are described as "intelligence sources" about those payments.

Did the Justice Department tell anyone in the White House about those payments? "As far as we can determine," a Justice Department spokesman said yesterday, there was no contact with anyone in the White House.

Attorney General Benjamin R. Civiletti has said that neither he nor anyone he knew of in his department had contacted the White House about the investigation into Billy Carter.

Unaddressed so far is the possibility that Justice Department investigators inadvertently tipped off Billy Carter in the course of their own inquiries into the "intelligence" information about the payments he had received from Libya. Conceivably, investigators following up on that information could have made inquiries that tipped their hand to Carter.

Whatever the provocation, the meetings Carter arranged on June 10 took place the next day. He went first to the Justice Department, then to the White House to see Brzezinski.

What happened at the Justice Department is unclear. Presumably, Carter was told that the department knew about the money he had received from Libya. According to Brzezinski, when he met with Carter and learned that he was again involved with the Justice Department, he quickly referred Carter to Lloyd Cutler, the president's lawyer.

Cutler, in turn, says he referred Carter to Washington attorneys Stephen Pollak and Henry Ruth, whom Carter then retained.

Both Brzezinski and Carter insist they did not learn from Billy Carter that they that he had received $220,000 from Libya. How these conversations could have occurred without that intelligence being exchanged without that intelligence being exchanged remains to be explained. Sources say this matter will be discussed in a White House statement on the affair that is currently being prepared for release in the near future, perhaps today.

The White House maintains that no one there learned about Carter's payments from Libya until July 11, a week ago Friday. Originally, it was expected that Carter's formal registration as a Libyan agent would be filed that day. In the end, it was delayed until Monday, July 14.

On tuesday night, July 15, Billy Carter appeared on the ABC News program, "Nightline." He was asked, "Have you talked to your brother about the Libyan flap?

"No, I have not. I have not talked to Jimmy about it at all," Billy replied. He was then asked if he intended to talk to his brother.

"No, I do not," he said.

Two days later, President Carter made his first statement on the affair to reporters in Florida. This is what he said.

"Just a few days ago, I recommended to Billy that he go ahead and make a complete revelation of what happened to the Justice Department. He thought it over for a while and then decided to do so. But I did not know about the activities before that."

Billy Carter may be asked to explain the discrepancy between his and his brother's statements when he appears this morning on ABC's "Good Morning America." The White House statement being drafted is also expected to address this apparent contradiction.

Why did Carter wait so long to register as an agent, and why did he finally do so last week? The only answer he has given to this question was on ABC last Tuesday night:

"I registered because of the investigation, and I was afraid of criminal charges or a grand jury. . . ."

Apparently these were new fears, born of the realization that after accepting $220,000, he could no longer plead that he was "just friends" with the Libyans.

But what was Carter's relationship with the Libyans? This fundamental mystery remains. Billy Carter insists that he never did anything for the Libyans except "show them some friendship."

According to the statement Carter filed with Justice, he "is not presently engaged in any activities on behalf of the Socialist People's Libyan Arab Jamahiriya [Libya], and presntly has no activities on its behalf under consideration."