For Mahmoud Abbas and U.S. Jewish leaders, their second date featured a little more substance and a little less flirtation. And this time the Palestinian Authority president brought a wing man.

Abbas and his prime minister, Salam Fayyad, met separately Tuesday evening with Jewish leaders in New York -- a sign of understanding on the Palestinian side of the importance of Jewish sensibilities, in Israel and the Diaspora, to advancing the peace process.

Abbas at the meeting seemed ready to move forward on some substantive issues, which took place during the launch of the U.N. General Assembly session.

In the first meeting, in June, Abbas had frustrated Jewish leaders by dodging issues of substance -- returning to direct talks and incitement -- but set a tone unprecedented in Palestinian-Jewish relations by recognizing a Jewish historical presence in the land of Israel.

When a group of Palestinian intellectuals challenged Abbas on the issue a month later, instead of backtracking -- typical of the one step forward, two steps back peace process tradition -- his envoy in Washington, Ma'en Areikat, repeated and reaffirmed the comments.

In the interim, direct talks have been launched, and Abbas was prepared to move forward on some substantive issues at Tuesday's meeting.

"I would like for us to engage in a dialogue where we listen to each other and where I can respond to your questions because I trust we have one mutual objective -- to achieve peace," he said, according to notes provided by the Center for Middle East Peace.

The center, a dovish group founded by diet magnate S. Daniel Abraham, sponsored the Abbas meeting, as it did in June. The Fayyad meeting was sponsored by The Israel Project, which tracks support for Israel in the United States and throughout the world.

Making his clearest statement to date on the matter, Abbas said he would not walk away from negotiations should Israeli Prime Minister Benjamin Netanyahu fail to extend a partial 10-month moratorium on settlement building set to lapse next week. The PA leader suggested that a way out might be if Netanyahu does not make a public issue of the end of the moratorium.

“I cannot say I will leave the negotiations, but it’s very difficult for me to resume talks if Prime Minister Netanyahu declares that he will continue his activity in the West Bank and Jerusalem," Abbas said.

Netanyahu is under pressure from the settlement movement not only to end the moratorium, but to resume building at levels unprecedented in his prime ministership. The Israeli leader also is heedful, however, of Obama administration demands that the parties not go out of their way to outrage each other.

Among the Jewish leaders at the Abbas meeting were Malcolm Hoenlein and Alan Solow, the executive vice chairman and chairman of the Conference of Presidents of Major American Jewish Organizations; Abraham Foxman, the Anti-Defamation League's national director; and leaders of umbrella groups such as the Jewish Council for Public Affairs and the Jewish Federations of North America.

Also on hand were Clinton administration foreign policy mavens such as Sandy Berger, Madeleine Albright and Daniel Kurtzer, who maintain close ties with Obama's foreign policy team.

Abbas also showed that he was attempting to bridge a gap on what until now seemed an intractable issue.

The Palestinians have long accepted the inevitability of a demilitarized state, but they reject a continued Israeli military presence. Netanyahu told Jewish leaders in a conference call Monday that he would trust no one but Israeli troops to preserve Israel's security on the West Bank's eastern border. At the meeting, Abbas floated the idea of a non-Israeli force that would include Jewish soldiers.

On other issues, Abbas was less prepared to come forward.

Israel wants a clear commitment from the Palestinians that any discussion of the refugee issue would clearly preclude a flooding of Israel with descendants of refugees of the 1948 war, which Israelis say is a recipe for the peaceful eradication of Israel. Behind closed doors, the Palestinians have said they are ready to provide Israel the assurances it needs, but Abbas said at the meeting only that it is a final-status issue.

Another issue could yet scuttle the talks now that the parties seem ready to put the settlement moratorium behind them.

Netanyahu, having extracted what seems to be an irreversible Palestinian recognition of Israel during his previous turn in the job, in 1998, now wants the Palestinians to recognize Israel as a Jewish state -- a result of the emergence of movements that seek to strip Israel of its Jewish character.

Abbas has resisted, in part because he sees such recognition as cutting off the 20 percent of Israel that is Arab, but also because he seems baffled by the demand. He argues that states are free to define themselves and should not need the approbation of others.

"If the Israeli people want to name themselves whatever they want, they are free to do so,” the PA president said.

In a sign that he also was seeking conciliation on the matter, Abbas said at the meeting that he would accept the designation if it were approved by the Knesset. He repeated his recognition of Israel's Jewish roots and decried Holocaust denial.

It was not far enough for some of his interlocutors.

Stephen Savitzky, the president of the Orthodox Union, wanted Abbas to recognize not only Jewish ties to the land but with the Temple Mount, the site of the third holiest mosque in Islam.

"President Abbas missed an opportunity this evening to make a key statement that would have created good will in the Jewish community," Savitzky said in a statement.

Fayyad, less charismatic but deemed more trustworthy than Abbas by the pro-Israel intelligentsia, appeared to fare well in the dinner hosted by The Israel Project, which hews to the centrist-right pro-Israel line of much of the U.S. Jewish establishment. He scored points for admitting that the Palestinian Authority had not done enough to combat incitement.

"Prime Minister Fayyad’s spirit of hope was extremely welcome," said Jennifer Laszlo Mizrahi, a founder of The Israel Project.

"We know that some people will criticize us for falling for a Palestinian ‘charm offensive.’ However, there is nothing offensive about charm. More Jews and Muslims, Israelis and Palestinians, should sit together over dinner and exchange ideas -- especially when it can help lead to security and peace.”