DFL Gov. Mark Dayton and his incoming lieutenant governor, GOP Sen. Michelle Fischbach, met for the first time Friday to discuss their unlikely partnership, agreeing they could find common ground on some issues — but not on the still-open question of whether Fischbach must give up her seat in the state Senate.

Over a lunch of Minnesota walleye and ice cream sundaes, Dayton and Fischbach talked about the unusual situation they’ll enter into next month, when current Lt. Gov. Tina Smith heads to Capitol Hill to replace U.S. Sen. Al Franken. Smith’s departure will trigger a succession clause in the state Constitution that requires Fischbach, the Senate president, to take over as the governor’s No. 2, even though she belongs to a different political party.

In a statement, the governor said Fischbach will be an “independent Constitutional officer” entitled to express her own views about state policies, though that “should not prevent us from having a constructive working relationship.”

Dayton said the question of whether Fischbach can simultaneously serve in both the executive and legislative branches came up, but he was not prepared to offer a definitive opinion.

“I told her that I am fully prepared to work with her as Lt. Governor; her standing in the Senate is beyond my purview, which is why I have requested an opinion from the Attorney General on that subject,” he said.

A spokesman for Attorney General Lori Swanson confirmed that Swanson has received the governor’s request to review the issue, but declined to say when the attorney general would be prepared to respond. Fischbach and Senate GOP leaders have said state law allows a member of the Legislature to serve as lieutenant governor, citing an 1898 court case on the matter. Senate DFLers contend that the Minnesota Constitution specifically prohibits such a situation, citing a 1970s amendment that they say superseded the earlier court decision.

In a separate statement, Fischbach maintained her position that she’ll serve as both lieutenant governor and as the senator representing her St. Cloud-area district.

“I look forward to fulfilling my constitutional duty of ascending to the role while maintaining my responsibilities to the constituents of Senate District 13,” she said.

Fischbach did not elaborate on other details of her meeting, other than that she and the governor “discussed ways to move forward” and, after sharing family photos, agreed that they both have “adorable” grandchildren. Dayton echoed those comments in his statement, and noted that the pair also discussed their pets: he has dogs and she has a cat.

The governor also noted that he and Fischbach are working to identify issues on which they could work together. He said she suggested elder care as one topic.

Meanwhile Friday, Smith announced that she will keep two of Franken’s top aides on her staff once she joins the Senate. Jeff Lomonaco will continue to serve as chief of staff, and Alana Petersen will serve as deputy chief of staff.