Mayor Sharon Pratt Dixon and D.C. Council Chairman John A. Wilson (D) left their first serious meeting on the city's budget crisis saying they had made a new vow Friday to cooperate with each other.

Whether Wilson's withdrawal of his budget proposal means they have produced a treaty -- or merely a truce -- won't be clear for some time. One month into their terms, and after a few weeks of political bickering, there's little doubt that the two are still sizing up each other, and figuring out how to share power.

The stakes are high -- for Dixon, Wilson and District residents. With a projected $300 million deficit forcing debate on new taxes, reductions in programs and furloughs of city workers, cooperation between the mayor and the council leader is crucial. Council members and other political observers say they expect little progress until Dixon and Wilson find a way to get along.

Both say they have the same goal -- pulling the city from the brink of bankruptcy -- but until now they have followed very different paths, and have displayed much different styles.

Wilson, a blunt, pragmatic politician and financial expert who has spent 16 years on the council, is averse to taking risks and has pursued options he knows he can control: taxes, spending cuts, furloughs. Meanwhile, Dixon, a lawyer and former utility company executive who had never held elected office, is betting that her behind-the-scenes lobbying on Capitol Hill will win the District at least $100 million in emergency aid.

Each won election with a large majority and each has strong views on how best to lead the District, long wracked by financial and administrative problems, in a new direction.

Though they work only three floors apart in the District Building, the two had not talked with each other much in recent weeks as they promoted their plans through the media. But Friday, Dixon and Wilson spent two hours huddled with D.C. Delegate Eleanor Holmes Norton (D) in a meeting arranged by Norton. And afterward, Wilson promised to hold his tongue on the deficit for a month and let Dixon work whatever magic she can in Congress.

"All of this is a major gamble," he said.

Dixon has avoided sparring with Wilson, and has said she believes they are communicating well. A high-ranking Dixon aide, who asked not to be identified, said Dixon does not intend to criticize Wilson.

"Battling back is not what she is going to do," the aide said. "That's not what she thinks the city wants."

The mayor's top priority in the first month has been recruiting members for her Cabinet and lobbying members of Congress for financial assistance -- which didn't leave a lot of time for cultivating council members.

Dixon has been troubled by criticism of her first month, especially in light of the mandate she claimed in her victory last fall.

Wilson has been disappointed with Dixon's start and has not been reluctant to say so. Last week, he criticized Dixon's handling of the budget crisis, saying that she was unrealistic in counting so heavily on Congress for assistance and that she had sent the council mixed signals on what she wanted to do.

His rebuke in part reflected the frustration that other council members say they have with Dixon so far. Though their expectations remain high, they are complaining that her first month at work has produced a series of missteps.

"I think she's doing about as well as you can expect for a new mayor," said council member William Lightfoot (I-At Large). "The problem is that for the council, moving from December to January is no quantum leap. For Sharon, it is."

One major point of dispute has been the council-approved gun-liability bill, which would allow shooting victims or their families to seek damages from manufacturers and dealers of assault weapons. Sensing strong opposition to the measure in Congress, Dixon sent word to Wilson early in January that her request for $100 million in aid was in jeopardy.

A private deal to repeal the bill was struck, but Dixon began voicing her concern publicly. That angered some council members, who said they now cannot avoid the appearance of trading legislation for money if they vote for the repeal.

Council members say they are annoyed with other Dixon moves. They include her decision not to send them proposed revisions in this year's budget and a proposed budget for fiscal 1992 until March 1 -- a month later than usual -- and her shifting position on whether the University of the District of Columbia's trustees should be dismissed. There also is confusion about how she intends to cut the city's bureaucracy.

Dixon has begun meeting with council members individually and is expected to appoint a liaison to them soon. Dixon administration sources said her choice is Ira Stohlman, a former council staff member.

Council member John Ray (D-At Large), who ran second to Dixon in last fall's Democratic mayoral primary, said she is at a disadvantage because she has had to balance choosing a staff and improving relations on Capitol Hill with solving an enormous cash shortage.

"The council doesn't know all that she stands for yet, so she is not as easy to embrace," Ray said.

Then there is Wilson, who is intent on having the council be an equal partner with Dixon. Council members and political allies describe him as stubborn and short-tempered, but say he genuinely believes that the city has no choice but to erase its deficit immediately.

Wilson critics say privately, though, that he is angling to seize more power than Dixon, and is not allowing her time to get her administration in order. Wilson dismisses those suggestions.

"I don't want to be mayor," he said in an interview last week. "They said the same thing about me all through the Barry administration too, and I didn't run against Marion -- six of his friends did."