Is it permissible to enter into a limited partnership for a specific period, with an agreement to take a variable percentage of the monthly profits based on the remaining capital, a portion of which is returned each month, so that the entire capital is recouped at the end of the period, along with an agreement not to guarantee the capital, knowing that one of the parties deposits money with the other?

It is permissible for a partner to enter with their existing goods in two ways: by selling specific, isolated goods, or by selling an undivided share of the shop's goods after inventorying and valuing them.

However, it is not permissible to return a portion of the capital monthly and give a fixed profit without inventorying. But if the intention is to divide the profit after inventorying and knowing the actual profit, then it is permissible.

There is no harm in saving money as a trust (deposit). However, if it involves using the money as a loan stipulated in the transaction, it is prohibited by the majority of scholars, though the Hanafis permitted it. If it is without stipulation, there is no harm.