Gov. Mark Dayton and Republican legislative leaders began meeting Thursday morning for their first court-ordered mediation session with a retired Hennepin County judge.

The Minnesota Supreme Court directed the DFL governor and GOP leaders to attempt to sort out a monthslong dispute over Dayton's veto of legislative funding. Both sides selected mediator Rick Solum, a former judge and trial attorney, and agreed to meet Thursday and Friday at the Dorsey & Whitney law firm in downtown Minneapolis, where Solum once worked.

Senate Majority Leader Paul Gazelka, R-Nisswa, posted on Twitter on Thursday morning that he was "Heading into mediation at 10."

"The results are important for MN's future," he added. "Hopeful."

Dayton's spokesman sent out an e-mail late Thursday saying that the day's session wrapped up at 6 p.m. and that it would continue Friday.

Further details about the mediation sessions, including the topics to be discussed and the people involved, have not been shared with the public. In addition to Gazelka, it's likely the process will involve the governor and Minnesota Management and Budget Commissioner Myron Frans, as well as House Majority Leader Kurt Daudt, R-Crown, attorneys for both sides, and other top legislators and staff members.

The sessions will cost taxpayers at least $1,088.25 in legal fees per hour, between the hourly fees for Solum ($257) and the two lead attorneys for the governor and the Legislature. The full cost of the dispute is still unknown, but by late August, the governor's office had been billed for $245,000 in legal fees. The Legislature will receive its own bill when the issue is resolved.

Nearly four months after the dispute began, the two sides remain divided on a number of issues. Amid a heated dispute over a tax-cut bill and a handful of policy issues, Dayton struck funding for the state House and Senate out of the budget to try to get GOP leaders to negotiate. The Legislature instead sued, arguing that the governor had violated the Minnesota Constitution by restricting the power of another branch of government.

A lower-court judge sided with the Legislature, but Dayton appealed to the Minnesota Supreme Court, which said the governor's veto was constitutional — but declined to formally rule on the matter until the two sides tried again to sort out their differences.

Dayton and legislative leaders enter into mediation with differing opinions on the veto itself, on whether or not a court can order emergency funding to keep the Legislature running, and on how much money the House and Senate actually need to keep their doors open. The Legislature has said it will have to furlough staff members and shut down operations at the Senate on Dec. 1 and the House on Feb. 1 if the governor's veto stands.

Both sides are expected to report back to the Supreme Court on their progress by the end of the month.

On Wednesday, Dayton declined to speculate about the mediation process, citing an agreement from both sides to keep the discussions private.

"I have nothing more to say about the mediation process, which begins tomorrow," he said. "If it's not successful, I will have a lot to say."