Gemara

(Rav Yehudah): "Lema'an Sefos ha'Ravah Es ha'Tzeme'ah" (joining the thirsty (desirous) and the satiated) refers to one who marries off his daughter to an old man... or returns an Aveidah to a Nochri.

Bava Kama 113b (Rav Chama bar Gurya): One may keep an Aveidah of a Nochri. We learn from "k'Chol Aveidas Achicha." You return an Aveidah of your brother, but not an Aveidah of a Nochri.

Question: Perhaps that only exempts one from taking the object to return it. What is the source that one who took it may keep it?

Answer (Ravina): "And you found it" connotes that he already picked it up (and still, you must return only to your brother).

(Beraisa - R. Pinchas ben Ya'ir): If Chilul Hash-m (desecration of Hash-m's name) will result, even keeping a Nochri's Aveidah is forbidden.

Rishonim

Rambam (Hilchos Gezeilah 11:3): One may keep the Aveidah of a Nochri. One must return "Aveidas Achicha." One who returns it transgresses, for he supports Resha'im.

Be'er ha'Golah (CM 266:2): This shows that the Isur does not apply to Nochrim nowadays, who admit to the Creator, and part of their ethics is to return Aveidos.

Rambam (ibid): If he returned it for Kidush Hash-m, so people will glorify Yisrael and know that they are trustworthy, this is praiseworthy.

Be'er ha'Golah (3): We learn from the case of Shimon ben Shetach in the Yerushalmi (Bava Metzia 8a) and Medrash Tehilim (12) MAtrona. (Once, Shimon ben Shetach's Talmidim bought a donkey for him from a Yishmaeli. A gem was hanging on it. He said that even though one may keep a Nochri's Aveidah, I do not crave wealth. To hear him bless the G-d of Yisrael is greater than all wealth of this world! (Many cases are brought in which Nochrim said this when Tzadikim returned Aveidos to them. Medrash Tehilim recounts a Chasid who returned a box full of coins to a prestigious Nochris. People said that he did not know that was inside; she said 'even the outside is gold!' She decreed that he can enter or leave without permission. He stopped people from engaging in trickery.

Rambam (ibid): If there will be Chilul Hash-m (if he keeps it), it is forbidden and he must return it.

Kesef Mishneh: Presumably, in such a case he must return it even if he did not yet pick it up, lest there be Chilul Hash-m. E.g. if a Nochri lost in a place of mostly Yisre'elim, if it is not returned, he will think that a Yisrael took (Beis Yosef - stole) it.

SMA (3): He explains that the Nochri will think that he did not drop it, rather, a Yisrael stole it from him.

Be'er ha'Golah (4): I think that the Beis Yosef should say 'they wil think that a Yisrael found it.

Rambam (ibid): We always take in their Kelim so that thieves will not take them, like we do for Yisre'elim, for Darchei Shalom.

Magid Mishneh: This is from the Yerushalmi.

Rosh (Bava Kama 5:6): Rabbeinu Meir mi'Rottenberg says that if a Nochri (Tony) lent to Reuven and took a security, and he lost it and Shimon found it, he returns it to Reuven, for Reuven owns it and Tony merely had a lien on it. Once he lost it and Shimon found it, his lien lapsed, for one may not return an Aveidah to a Nochri. If Shimon wants to return it for Kidush Hash-m, let him do so with his own money, and not with Reuven's money without Reuven's consent!

Poskim

Shulchan Aruch (CM 72:38): If Tony lent to Reuven and took a security, and he lost it and Shimon found it, he returns it to Reuven.

Bach (31): According to Rabbeinu Meir, Tony's lien lapses only after Shimon finds it, because Shimon may not return it. However, the Re'em (in Mordechai, Chezkas ha'Batim) holds that his lien lapses once he loses it, and Reuven acquires it wherever it is. If so, if Shimon gave it to Tony or any other person, he must pay Reuven, like a damager. Rabbeinu Meir holds that if he returned it to Tony he transgressed returning an Aveidah to a Nochri, but he need not pay.

Rebuttal (Shach 152): Also Rabbeinu Meir obligates him to pay.

Poskim

Shulchan Aruch (CM 266:1): One may keep the Aveidah of a Nochri. One must return "Aveidas Achicha." One who returns it transgresses, for he supports Resha'im. If he returned it for Kidush Hash-m, so people will glorify Yisrael and know that they are trustworthy, this is praiseworthy.

Beis Yosef (DH Aveidas): Surely we do not distinguish between Nochrim who serve idolatry and those who don't, for none of them are Achicha. The Tur says that we do not return to idolaters. This is not for a Diyuk (that we return to Nochrim who do not do idolatry). Perhaps he said so because in Eretz Edom, Minim used to besmirch Yisrael in the eyes of kings due to this and similar laws. Chachmei Yisrael responded 'these laws applied only to Nochrim the days of the Gemara, who served images and did not admit to the Creator. They were called idolaters. These laws do not apply to Nochrim nowadays, who recognize the Creator.' Alternatively, the Tur said 'idolaters' to include a Yisrael idolater. His Aveidah is permitted, like a Nochri's.

Beis Yosef (DH u'Mah she'Chosav v'Lo): Rashi explains that one may not return to a Nochri, for this shows that he equates Nochrim to Yisrael, and he does not consider returning Aveidaos to be Hash-m's Mitzvah, for he returns to Nochrim, even though he was not commanded.

SMA (2): Rashi explains that "thirsty" refers to Yisraelim, who desire to serve Hash-m, and "satiated" refers to Nochrim, who have no desire to serve Him. The Rambam explains like the Targum, that Yisrael adds Shogeg Aveiros to Mezid. A Yisrael who returns an Aveidah to a Nochri sins b'Shogeg, for he supports sinners.

SMA (4): The Torah forbids only when the Nochri does not know that it fell or who found it, but not if he intentionally left it in his Chatzer and did not fathom that it would be lost from there. If the Yisrael knows that thieves came and will take it, since it is not considered an Aveidah, the Yisrael may take it to a place guarded from the thieves, for Darchei Shalom.

Pischei Teshuvah (1, citing R. Akiva Eiger Bava Metzia 22b): If Reuven was a Shomer for a Nochri Joe, and the item was lost, the finder must return it. Even though the item itself belongs to Joe, he returns it to save Reuven from a loss, i.e. the need to pay for it. This is not only like R. Shimon, who obligates for Davar ha'Gorem l'Mamon (something without value, but it can save someone from a loss). Rather, it is even like Chachamim. However, Moreinu ha'Rav Yisrael says that Chachamim would exempt, for it is not Aveidas Achicha. I bring a proof from the Pnei Yehoshua's question (27a DH bi'Gemara). Why did the Gemara ask 'how does the other opinion learn Ravina's law?' Perhaps he holds like the opinion that permits Gezel Nochri. If so, we need no verse to exempt from returning his Aveidah! I say that this case (of the Shomer) it is not called Aveidas Acicha, and we need the verse. One may not steal the item, because a Yisrael is responsible for it. In practice, this requires investigation.

Rema (388:12): If Shimon wanted to flee and not pay to Nochrim what he owed them, and Reuven told them, he is not a Moser (informer), for he only causes Shimon to pay what he owes. However, this is a great evil, for it is like returning an Aveidah to a Nochri. If this caused damage, he must pay the damage.

SMA (31 and Shach 61): He must pay only if he caused damage above what Shimon owed.

Gra (77): If a Nochri erred, one may keep it, for this is like his Aveidah.

Question (Be'er ha'Golah 60): Rabbeinu Meir himself wrote that if a Nochri (Joe) deposited or lent to David, and Joe died, and David does not return it to the heirs when the claim it, David sins! This is unlike an Aveidah, which the Torah permitted. It is good to return even an Aveidah when it is Kidush Hash-m! The custom of Kehilos Yisrael, who are Tzadikim and straight, do not do any Sheker or injustice to Nochrim. We announce and allow publicizing to Nochrim people who buy on credit or borrow without intent to pay.

Gilyon Maharsha: Tosfos (Bava Kama 114a DH v'Lo) says that if Reuven testified about what Shimon owed to Nochrim, we do not excommunicate him. The Rosh (10:14) permits to testify l'Chatchilah (as long as they will not obligate Shimon unlike Torah law).