PHOENIX — Sen. John McCain’s death in office has handed Arizona’s governor an empty Senate seat to give out — and a difficult political puzzle to solve before he does.

Arizona law requires only that Gov. Doug Ducey name a replacement who is a member of McCain’s Republican Party and who will fill the seat until the next general election in 2020. But in a state with a deeply divided Republican Party, where McCain was a towering but divisive figure, the choice is far more complicated.

Ducey is balancing the demands of the many conservative Arizona Republicans who have soured on McCain due to his dovish immigration stance, criticism of President Donald Trump and vote against a rollback of President Barack Obama’s health care law. They are wary of Ducey appointing a moderate. But naming someone with dramatically different views from McCain could be viewed as disrespectful to McCain’s legacy, carrying its own risks.

In either case, Ducey wants to set the party up to hold the seat two years from now, no easy task given the turmoil in his party.

The decision is under scrutiny in Washington. While McCain has been treated for cancer in Arizona and unable to vote in Washington, his party’s already narrow Senate majority had shrunk from two votes to one. With the confirmation of Trump’s Supreme Court nominee, Brett Kavanaugh, scheduled for next month, the GOP needs every reliable vote it can get. Ducey’s office has heard from Vice President Mike Pence’s aides about the choice, a person familiar with the discussions said Sunday.

A day after McCain’s death, political types from Arizona to Washington were buzzing with options. The senator’s wife, Cindy McCain, was viewed as a possibility, as was former Sen. John Kyl and former McCain chief of staff, Grant Woods. Former lawmakers and state officials were floated as middle-ground options, including Ducey’s chief of staff Kirk Adams, who might not anger the right wing of the party.

“If he picks someone too far left, we’re going to have a primary in two years,” said Constantine Querard, a conservative Republican strategist.

Ducey faces a weak primary challenge from his right in the state’s primary elections Tuesday, and spokesman Daniel Ruiz said Sunday that the appointment will not be made until after McCain’s funeral, which will likely be next week.

Doug Cole, a former McCain staffer and veteran Arizona strategist, said one of Ducey’s key choices has to be whether he wants to name someone who wants the job for the long term. “Do I appoint a caretaker or do I appoint someone who will stand for election?” Cole asked.

The person who was previously seen as McCain’s most likely successor is Rep. Martha McSally. Like the late senator, she’s a former fighter pilot — one of the first women to fly in combat and an Air Force colonel. But she is running for the seat vacated by Sen. Jeff Flake, who, like McCain, outraged the state’s conservative base for bucking Trump on immigration and other issues.