Is it permissible to possess part of an endowed land, given that the Awqaf Authority stipulated the construction of a two-story building upon it in exchange for this possession, with the remaining part retaining its endowment status, especially since the endower did not stipulate this?

The endowment must remain intact and be managed according to the conditions of the endower. It is not permissible to sell it or dispose of it in a way that removes it from its endowed status. This is the fundamental principle, based on the Prophet's (peace be upon him) saying: "If you wish, you can detain its origin and give away its fruit [in charity]." So Umar gave it away in charity, [stipulating] that its origin should not be sold, inherited, or gifted. Imam Malik, Imam Shafi'i, and Imam Ahmad, in one of the narrations from him, held the view that it is absolutely impermissible to sell or replace an endowment. Imam Ahmad, in another narration, held the view that it is permissible to sell or replace it if its benefits have been completely lost or if there is a clear public interest in doing so. He based this on the action of Umar (may Allah be pleased with him) when he moved the treasury. This latter view is also that of Sheikh al-Islam Ibn Taymiyyah, who even went further to permit the exchange of an endowment for a better one if there is a clear public interest.

Therefore, there is no harm in purchasing an endowed land from the authority responsible for endowments if that land is unutilized and cannot be exploited for the benefit of the endowment except by selling it or selling a portion of it.