In Iceland, the president runs for office but never talks about politics.

That alone would qualify Iceland as an unusual country -- leaving out other notable facts like:

* All-day sun on the summer solstice in the northern part of the country;

* Year-round outdoor swimming due to warm underground springs;

* One of the highest per capita incomes among European countries, with even distribution;

* A population of 230,000 people and, in the fall, 2 million sheep. ("Iceland without sheep wouldn't be Iceland," says the president.)

"The post of president is absolutely unpolitical," says President Vigdis Finnbogadottir, 52, whose surname means "daughter of Finnbogi," her father having been Finnbogi Rutur Thorvaldsson. "Somebody has to be above these political things . . . Politics are rather emotional."

When she meets with President Reagan tomorrow, they will talk about cultural relations between the two countries among other things, "because President Reagan is a cultured man," she says. But there will be no politics, she says adamantly. "The president never mentions politics," she says. "The president is never asked."

The president of Iceland is an extraordinary statistic herself. She is the only democratically elected female head of state, according to Icelandic officials. She possesses handsome Nordic looks, and she is called by her first name, as are most Icelanders. Exceptions to this rule include some religious leaders and the prime minister, who runs the Althing, the Icelandic parliamentary government, and is not above politics.

"I see written, 'President Finnbogadottir,' and it seems so strange," she says, sitting in her suite at the Madison Hotel. "It doesn't exist." In Iceland, surnames -- denoting daughter or son of someone -- mean nothing. In the phone book, full names are listed alphabetically by first name.

Vigdis will join other Scandinavian officials in opening the "Scandinavia Today" exhibition, which begins here tonight with a fireworks display choreographed to music on the Mall and will continue for several months with exhibits, films, concerts and lectures on the arts of Denmark, Finland, Iceland, Norway and Sweden. Vigdis will deliver the keynote address tomorrow afternoon at the Kennedy Center. She will be here all week on a working visit before going to Minneapolis, New York and Seattle, other cities where "Scandinavia Today" will be held. (Los Angeles and Chicago are also hosting exhibitions.)

She was approached about running for president in 1980. At the time she was director of the Reykjavik Theatre Company. "You see, in Iceland, you do not wake up one morning and think, 'I want to become president,' " Vigdis says. "The tradition is that someone comes to you and asks you to stand . . . They see something in the person whom they want to represent the country. I said in the campaign that I was not standing for the country. I was standing for the people who wanted me. I was the spokesman."

She says quite clearly that some people had made a conscious decision to search out a woman candidate. "There were people who thought it was time that a woman should stand," Vigdis says, "so they started to look for a lady. Many names were mentioned. But I was finally pushed. I don't know what they saw in me."

The daughter of a civil engineer and professor at the University of Iceland and a nurse who was chairman of the Icelandic Nurses Association, Vigdis had become known for her lectures on Icelandic culture both in Iceland and abroad.

If voters had hesitations about her being a single, divorced woman with a young daughter, Vigdis sought to dispel them: "If a man is president and he loses his wife, would you ask him to resign?" she asks. "Everybody can become single, again."

She won the election on June 30, 1980. She is the fourth president of the Republic of Iceland, which declared its full independence from Denmark in 1944.

"When I was elected, it was news all over the world," she says. "I received newspaper cuttings in Chinese with all kinds of letters in languages I didn't understand."

The peculiarities of her rule have much to do with the smallness of Iceland -- a country with no armed forces save a coast guard and a government so compact that the foreign ministry takes up one floor of a government office building in the capital city of Reykjavik. "But a large floor," says Vigdis. "It's not two rooms and a kitchen."

"It's definitely one of the smallest foreign ministries," says Olafur Egilsson, chief of protocol, who has accompanied Vigdis on this trip, "but it's larger than the State Department in Thomas Jefferson's time."

What does the president do?

"I can't say I sit in a deep chair all day," says Vigdis, mulling over the question and fingering a pair of pinkish-brown-framed eyeglasses that she holds. "I always find it difficult to describe what I'm doing all day. Gentlemen," she says, turning to the two Icelandic officials sitting with her, "what do I do all day? I receive quite a lot of people -- people who have different causes . . . Anybody in Iceland can make an appointment to see the president."

And one wouldn't have to wait too long. "If you invented a new boat-safeguarding measure and you wanted me to introduce it in my next speech, I think I would see you this afternoon." In any case, two or three days a week Vigdis spends the afternoons receiving people.

"We have very deep roots," she says of Icelanders. "I have not yet met an Icelander abroad who is not a little homesick. Part of it is this island and part is this language. We are only 230,000 people who speak it . . . Every single person is needed to run the country."

Their problems are not unemployment -- they have none, says Vigdis -- and poverty. Instead, they worry about the declining supply of fish, their main export, and fiercely guard their fishing limits, which has twice pitted Icelanders against British fishing vessels in hostilities referred to as the "cod wars."

The image of Iceland as simply a glacial rendezvous point for presidents and chess champions is a faded one. Vigdis points out that student exchanges between the United States and Iceland flourish and fuel enthusiasm for both countries. "Since we are a steppingstone between America and Iceland," Vigdis says, "we are cosmopolitan. We have theaters and television and radio. We even have nightclubs and discos. We have everything, only on a small scale."

As president, Vigdis spends much of her time traveling extensively throughout Iceland talking to citizens. She also attends openings of museums and theaters and meets with the prime minister. "He keeps me informed about what is happening -- but that's all," she says.

She signs bills and has the power to veto them, although exercising that right is unheard of. "It's first and foremost the democratically elected members of Parliament who make the bills," Vigdis says. "If, for some reason, the bill is vetoed, it goes back to the people for a national vote."

When she has time, she slips out of her office and goes to the theater she used to run. "Even a president has a coffee break in the morning," Vigdis says. "It's a three-minute walk to the theater, and since I am a good friend, they let me sit and watch rehearsals."

She studied French at the University of Grenoble and at the Sorbonne in the late '40s and early '50s -- specializing in drama -- before coming back to Iceland where she taught French in junior college and on television and was a tour guide. She was always interested in drama. "I was very enthusiastic about avant-garde theater," she says. "I was in the first experimental theater group in Iceland." She was director of the theater company from 1972 to 1980, when she left to run for president.

She put much effort into seeking out and encouraging new Icelandic playwrights. "The language we speak is the oldest in the region," says Vigdis, who speaks Icelandic, French and English. "Icelandic is the same as the old Norse spoken 1,100 years ago. Much is written in Icelandic because the word has always been so inspiring . . . People didn't have so many materials to make things out of. We didn't have wood. We don't have trees in Iceland. The lava has meant there are no good materials with which to make beautiful churches like Notre Dame. Iceland has many active volcanoes. That's why we have quite a lot of writers. It was a lot of fun finding writers."

She never wanted to be an actress. "Somehow I never wanted to stand on the stage," she says, "which is a very funny thing, since I am standing on a stage now."