We study stationary solutions of a one-dimensional low-Mach-number model derived in Gasser and Struckmeier (Math. Meth. Appl. Sci. 2002; 25(14): 1231) to describe fire events in long tunnels. The existence of solutions of the corresponding stationary model is shown to be equivalent to the existence of solutions of an algebraic problem. Multiple solutions are shown to be possible. The relation between different formulations of the problem is analysed. Weak and special distributional solutions are considered. Finally, numerical examples of realistic tunnel data with single and multiple solutions of the stationary problem are given.