We study a problem of exploitation of a groundwater resource, mainly used for irrigation, in which a water agency is needed in order to manage an exceptional and priority extraction of water for an alternative/new use (e.g. domestic water). To this goal, we build a two-stage discrete Stackelberg game in which the leader (the water agency) just intervenes when the new use takes place (in the second stage) and the follower is a representative agent of the regular users of the aquifer, i.e. the agricultural users. We study two types of Stackelberg equilibrium, which can arise depending on the agents' commitment behavior, namely openloop commitment) equilibrium and feedback (non-commitment) equilibrium. We analyze and compare extraction behaviors of the different agents for the different equilibria and the consequences of these extraction behaviors for the final state of the resource and the agents'profits. For some hypotheses on the parameters, theoretical results show that commitment strategies lead to higher stock levels than non-commitment strategies when the leader's weight assigned to the profits from the agricultural use is lower or equal than the one assigned to the profits from the non-agricultural use. However, performing numerical simulations relaxing previous economic assumptions, we show that there are situations in which non-commitment strategies could be more favorable than commitment strategies not only in terms of final stock of the resource but also in terms of users' profits.