moral

Thank you. Does that mean I'm supposed to fight over the water? Or should I cast lots for who drinks?

You said there's a reference in Talmud where it is?

In the Talmud (B”M 62 A”A) there is a famous discussion about a situation where there are owners of water. I thought you were talking about it. Regarding partnership, there is no discussion there, but the commentators talked about it. It seems that it can be said that the entire dispute is when the water belongs to one of them. Therefore, it seems that when the water belongs to both, even the Rabbis agree with Ben Petura that it is divided. But in my opinion, this is not reasonable, and they should draw lots. Although it is possible to make it difficult for me to explain why they did not present the dispute also in the case of shared water. This can be answered in several ways. Here are two examples: 1. They wanted to teach that in a situation where the water belongs to one, it is he who receives it and not by lot. 2. They wanted to teach that even in such a situation, both of them will drink according to Ben Petura (may the power of Ben Petura be heard).
See my article on separating Siamese twins here on the site.

I understand, thank you very much
Regarding Siamese twins, I think there is a difference. In Siamese twins, performing the action that will prolong the life of one of them will kill the other with his own hands, compared to the matter with the water, which indeed he died because I drank it and not him, but it is a projection that is not binding.

I did not compare it to the case of Siamese twins. I referred you to the article to see such a consideration. I distinguished there between a situation in which the common organ belongs to both of them and a situation in which it naturally belongs to one of them.
Incidentally, now I think that perhaps Ben Patura, who said that they would both drink, did not really mean that they would share, but that they would draw lots. His argument is that ownership of the water is of no importance (contrary to the position of the Rabbi) because I cannot drink and see my friend die just because the water is mine. But he also does not say that they would both drink and die, but that both have the possibility of drinking and being saved.
Although the wording there says “it is better that they both drink and die”, and this means that he really means that they will drink and die. But perhaps it can be argued that it is better that they both drink and die rather than that he drinks alone. But the real alternative is to draw lots. This is a bit narrow also because the opinion of the Rabbi is presented after Ben Patura. The Gemara should have brought the words of the Re, and then Ben Petura responds and says that if they say that the owner of the water should drink, then it is better that they both drink and die. Therefore, we should draw lots.
Well, just a thought that occurred to me now.

But apparently, if the lottery is good, then owning the water is an equally good lottery win?

Absolutely not. A lottery gives an equal chance to both, and therefore both will agree to it. Such a lottery is a realization of both drinking. In another formulation, we can say that now that we have divided the water according to Ben Patura's opinion, we agree to hold a lottery and whoever wins will receive the entire sum. Instead of dividing the water, we divide the chance of winning it. Each of the two will agree to this, because instead of receiving a temporary life, he receives a 50% chance of eternal life. Just as I described in the article on Siamese twins.
The distinction between ownership, which is a situation that already exists when the problem arose, and the lottery that is being held now, is also discussed in column 538.

And both of them fundamentally disagree, from the moment of their birth behind the veil of ignorance, with the rule that whoever owns a pitcher of water will drink it?

Unlikely. This is a hypothetical case. It does not detract from the evidence.

Another way of putting it: To the same extent, both would also agree behind the veil of ignorance that the water would be given to someone who does not own it. Both would also agree that a lottery would be held in real time. So why do you specifically choose the agreement that the water would be given to the owner of the water? Because the power is in his hands. We have returned from the lottery to power. See the above columns.

Since Ben Petura also gave a reason for his words, it is impossible to say that he also agreed to the lottery in principle, since his reason is that no one should see the death of his brother, and I am not permitted to agree to a lottery that will keep my friend alive and perhaps kill me, because then my friend will see my death, and this contradicts Ben Petura's principle, which he himself relies on the verses that explain that I am not allowed to leave my brother to die.

And it can be said that with this way of thinking, then it is possible to argue what Rabbi Akiva's opinion is in the case of partnership. As you can see, Rabbi Akiva did not dispute Ben Petura's moral assumption that it is better not to vomit up my dead brother. Rather, he disputed that this moral argument does not apply where my right to life (my ownership) takes precedence over his, and then I have no obligation to lose it for the right to his life (the sermon of "And live with me," that I should take care of him living with me but not lose my life for it). But where the right to life is not preferable to that of my friend, then the moral argument applies that it is better not to witness the death of my friend. Ben Petura is not interested in the fact that it is better for one to die than two, but rather he is concerned with the morality of not one living and seeing the death of his brother. Of course, this is just a thought. I would love to hear your opinion?

If the problem is to find some kind of priority for choosing ownership, then that seems like a small problem to me. Priority to the one who owns the kithon gives everyone the opportunity to prepare for the situation in advance and trust themselves (as most reasonable people want). In a situation of a lottery or transferring the kithon to someone who is not the owner, he actually has to check from the start that all the people in his group have water and are not wasting it. This is a much heavier burden. A more logical division of the burden is for everyone to take care of themselves and that way, in the end, everyone will benefit the most. Therefore, behind the curtain of ignorance, it is likely that everyone will agree to this. Therefore, it seems to me that if Ben Patura is satisfied with the lottery, then it is not necessary.

I don't understand the question. I explained it. Ben Patura answers the Ra that one should fear the death of his friend, meaning that the owner of the water should not drink and fear the death of the other. But the lottery is a different matter, since each of them is given a chance to live.
I didn't understand your second comment either. You repeat what I said. This is what I wrote in the simplest form of the question, that since they share in the case of ownership, it means that when there is no ownership, the Ra also acknowledges Ben Patura. I later rejected it.
I'm not sure I understood your third comment. Does it refer to the implied lottery that was proposed here? According to your proposal, this is not a lottery but a different consideration. If you had proposed a lottery or an agreement on ownership, I don't know why everyone would have preferred to agree. Therefore, in practice, it seems that you are actually using considerations of power in disguise here.