Having visited Italy and Ireland already this year, Hillary Rodham Clinton touched down today in the third "I" on New York's traditional ethno-political map, sweeping into Israel with a packed two-day itinerary.

On paper at least, Clinton's visit to Israel, her fourth as first lady, is an official trip that has nothing to do with her probable Senate candidacy in New York next year.

Her schedule, much of it as the guest of Nava Barak, wife of the Israeli prime minister, is crammed with speeches, functions and tours--more or less standard fare for an active first lady's visit to the Holy Land. Neither the address she delivered at Tel Aviv University today on combating youth violence, nor the speech on Middle East peace she is to give Thursday commemorating the birthday of the late prime minister Yitzhak Rabin, is overtly political.

Yet with the election for the Senate seat she is expected to seek less than a year away, much of what Clinton does is presumed to be political. And whatever her intention, there was no escaping the buzz of partisan politics audible even before her arrival.

The commotion started last week with a suggestion in the Israel media--almost certainly false--that Clinton had declined Nava Barak's invitation to visit Jerusalem's Western Wall, Judaism's most sacred spot. That item swiftly found its way into New York's tabloid press, prompting scathing commentary from small right-wing Jewish groups that already regard Clinton with barely concealed contempt.

And today New York City Mayor Rudolph Giuliani, the presumed Republican candidate, pointed out that he, too, has been to Israel--twice. "Both times I was there I went to the West Wall," he said.

True, as of last Friday, Clinton did not plan to go to the wall, which she visited in 1994. But amid the fuss in New York, her handlers found room for it on her Thursday schedule.

"Every move she makes is subject to very intense political calculus and also intense political speculation," said a sympathetic Israeli official. "She's damned if she does and damned if she doesn't."

Likewise, the minor tempest over the Clinton schedule's omission of plans to visit Palestinian-controlled territories. Lest that be interpreted as a snub, her staff squeezed in a lightning visit to the West Bank city of Ramallah. There, as the guest of Palestinian first lady Suha Arafat, she will present a U.S. government grant for refurbishing mother-and-child health clinics.

New York has a sizable Jewish electorate and Clinton has already found that the the Middle East is a political hot potato.

Last year, when she expressed her support for an independent Palestinian state, some Jewish groups were incensed. This year, when she embraced the Israeli position that Jerusalem should be the united and undivided capital of Israel, and that the U.S. Embassy should be moved there from Tel Aviv, she was accused of pandering to Jewish voters.

Both positions went beyond U.S. policy, which holds that Palestinian statehood and the status of Jerusalem are matters for the two sides to negotiate.

Beyond politics, Clinton's trip here is heavily concentrated on children's, women's and health care issues, all of which are among her long-standing interests. Accompanied by Nava Barak, she toured a counseling center for troubled youngsters today, spoke on fighting youth violence and was the guest of honor at a dinner this evening hosted by Israel's first couple.

Clinton's speech touched a nerve in Israel, where a number of sensational killings this year have grabbed newspaper headlines and prompted Nava Barak to lead a national campaign against youth violence. As she has in the past, Clinton stressed the responsibility of the entertainment industry for inspiring acts of violence.

Clinton is to deliver another speech Thursday, this one on Middle East peace. In addition to the planned visits to the Western Wall and to Ramallah, she is also scheduled to visit Yad Vashem--Israel's main Holocaust museum, a pediatric clinic and a women's health center. Her schedule then takes her to Jordan.

CAPTION: Hillary Clinton supports Israeli campaign against school violence during a visit being scrutinized for her political plans.