Can someone living outside of Israel ask a visitor from Israel to do a melacha for them on the second day of Yom Tov if the visiting Israeli is only keeping one day of Yom Tov?
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The main source is the shulchan aruch, 496:3, and commentaries, where a ben eretz yisroel who is traveling in chu"l is forbidden to do melacha.
The Ta"z permits such a person to do melacha in private, only. Most other poskim (including the magen avraham and the mishna brurah) forbid even this. The shulchan aruch harav is also strict on not doing any melacha (although his reasoning is kabbalistic in nature)
Many Israelis, however, do melacha in public. I have not found a source permitting this, however, I'm reluctant to call "everyone" wrong. I have been told (third-hand) that R' Goren permitted this, but I have not seen this in any of his writings (although I admit that I haven't looked).
If the Israeli is in an area where there are no other (non-Israeli) Jews, however, he may do melacha, even publically. Similarly, a traveler who leaves E"Y on Yom Tov Sheni
Note that the opposite case, a visitor from chu"l who is keeping two days who wants to ask his Israeli host to do melacha, has a different halacha... but that should get its own question!