The Minnesota Supreme Court’s ruling in a case pitting Gov. Mark Dayton against the Republican-controlled Legislature threw the dispute back to the state’s politicians with a message that might be summarized this way: Not our problem. Fix it yourselves.

“I think the court is clear: They expect the politicians to figure this out,” said Mary Jane Morrison, emerita professor of law at Mitchell Hamline School of Law and author of a book on the Minnesota Constitution.

The stakes are high: Without a resolution, the House and Senate will soon run out of money to pay legislators and their employees, following Dayton’s veto of their budget. In its ruling issued at the end of last week, the Supreme Court declined to strike down Dayton’s veto, but also suggested that it led to an “unconstitutional result.”

Instead of a definitive ruling, the court in the order signed by Chief Justice Lorie Skjerven Gildea directed both parties to hire a mediator by Tuesday and demanded a report on the status of the Legislature’s bank accounts by Friday.

Whoever ends up as mediator will have their work cut out for them. Rank-and-file Republican lawmakers remain staunchly opposed to a deal with Dayton, who is demanding that the Legislature reconsider some tax breaks and a few policy items from the legislative session earlier this year.

Dayton and legislative leaders stayed mum Monday, declining to comment on potential mediators and referring to statements released after Friday’s ruling, in which both sides claimed some vindication. A statement from Keith Hovis, spokesman for Dayton’s Minnesota Management and Budget agency, said lawyers for Dayton’s office and the Legislature have been in touch.

“We will have more information in the coming days surrounding mediation and next steps,” Hovis said.

By ordering mediation, the court’s decision treats Dayton and the Legislature like fighting spouses.

Mediation is one of the methods known as “alternative dispute resolution” that seeks to solve conflicts so a court doesn’t have to. It has become increasingly popular in Minnesota and across the country.

“It’s a process in which people who are in a conflict can come together with the assistance of someone not involved who can structure the conversation. The idea is they collectively figure out how to move forward,” said Sharon Press, law professor and director of Mitchell Hamline’s Dispute Resolution Institute.

Unlike arbitration, which is another way to resolve disputes without a judge, the mediator won’t make a final, binding decision. The parties will have to agree to whatever deal they make.

John Ella, a Minneapolis employment attorney who estimates that he has had 100 cases undergo mediation, said the process can be “amazing when it works well, and it usually works.”

He said usually the two parties are in separate conference rooms, with the mediator shuttling between the two rooms to first hear concerns and build trust and then try to strike a settlement that both sides can agree to.

Press said that in this case, she hopes the mediator brings the two sides together in the same room and forces them to work it out.

“I don’t think shuttle diplomacy is the way to go here,” Press said, referring to the practice of a mediator traveling back and forth between the two camps. “They need to figure out how to work together, talk together, do the work of the people of Minnesota, and if they can’t figure that out, nothing changes.”

The entire dispute dates to the final days of this year’s legislative session, in May. As Dayton and GOP leaders negotiated the state’s next two-year budget, Republicans seeking to get the governor to sign off on their tax-cut package — which they knew he disliked — linked funding for the state Department of Revenue to passage of the tax bill. In response, Dayton vetoed legislative funding, arguing that it was the only way to get Republicans back to the negotiating table.

The Legislature sued, arguing that Dayton violated the constitutional separation between equal branches of government. The two sides later agreed to a tentative deal that funds the House and Senate through Oct. 1.

Dayton wants to renegotiate five separate items that were tucked inside bills that he signed: He wants Republicans to scale back tax cuts, especially on wealthy estates, business property and smokers; repeal a new law blocking driver’s licenses for undocumented immigrants; and change another new law that makes changes to Minnesota’s system of teacher licensure.

Adding to the complexity of any attempt at mediation: Legislative leaders can agree to a deal via mediation, but they don’t have the final say on what happens at the Legislature. Ultimately, it’s the 201 lawmakers who will have to give any stamp of approval.

As for now, rank-and-file Republicans have shown little interest in negotiating with Dayton. They would rather give up their pay than bow to Dayton’s demands, two of them said Monday.

“That’s extortion. That’s a horrible precedent,” said veteran Rep. Matt Dean, R-Dellwood, a 2018 candidate for governor.

Press, the mediation expert at Mitchell Hamline, said there have been disputes that seemed far more intractable than this one but were resolved. She cited former U.S. Sen. George Mitchell’s work brokering peace in Northern Ireland.

“It’s the absence of hope that is really the killer,” she said.