Is the aforementioned contract permissible, in which a partnership machine was purchased by three individuals, and it is leased out with the condition that the lessee be notified two months prior to sale or repossession, and how should the rental income be divided among the partners, and does the absence of a specified contract duration render it impermissible?

As for the partnership contract in the machine's capital, there is no harm in it. The profit will be distributed among you according to your agreement, or failing that, according to each of your capital contributions. As for the contract for renting the machine monthly without specifying a time period, it is valid according to many scholars. The condition of giving a grace period to the second party before terminating the contract is permissible.