While single-level Nash equilibrium problems are quite well understood nowadays,less is known about multi-leader multi-follower games. However, these have importantapplications, e.g., in the analysis of electricity and gas markets, where often a limitednumber of firms interacts on various subsequent markets. In this paper, we consider aspecial class of two-level multi-leader multi-follower games that can be applied, e.g.,to model strategic booking decisions in the European entry-exit gas market. For thisnontrivial class of games, we develop a solution algorithm that is able to compute thecomplete set of Nash equilibria instead of just individual solutions or a bigger set ofstationary points. Additionally, we prove that for this class of games, the solution set isfinite and provide examples for instances without any Nash equilibria in pure strategies.We apply the algorithm to a case study in which we compute strategic booking andnomination decisions in a model of the European entry-exit gas market system. Finally,we use our algorithm to provide a publicly available test library for the consideredclass of multi-leader multi-follower games. This library contains problem instanceswith different economic and mathematical properties so that other researchers in thefield can test and benchmark newly developed methods for this challenging class ofproblems.