On the structure of the solution set of a generalized Euler-Lambert equation

The transcendental equation xex=z and its solutions, described by the Lambert W function, often occur in physics and mathematics. In the last decades it turned out that the study of similar but more general equations is necessary in molecular physics, in the theory of general relativity and also in the description of Bose-Fermi mixtures as well as in some combinatorial problems. In this paper we offer a full description for the solution set of a one parameter generalization of the above mentioned equation.