Quartic monoid surfaces with maximum number of lines

In 1884 the German mathematician Karl Rohn published a substantial paper on \cite{ROH} on the properties of quartic surfaces with triple points, proving (among many other things) that the maximum number of lines contained in a quartic monoid surface is $31$. In this paper we study in details this class of surfaces. We prove that there exists an open subset $A \subseteq \mathbb{P}^1_K$ ($K$ is a characteristic zero field) that parametrizes (up to a projectivity) all the quartic monoid surfaces with $31$ lines; then we study the action of $\mathrm{PGL}(4,K)$ on these surfaces, we show that the stabiliser of each of them is a group isomorphic to $S_3$ except for one surface of the family, whose stabiliser is a group isomorphic to $S_3 \times C_3$. Finally we show that the $j$-invariant allows one to decide, also in this situation, when two elements of $A$ give the same surface up to a projectivity. To get our results, several computational tools, available in computer algebra systems, are used.