Description of singularities for system “billiard in an ellipse”

A “billiard within an ellipse” is an integrable system appearing in the description of a point motion inside an ellipse with natural reflections at the boundary. This system is considered in the paper, the topological invariant of Liouville equivalence of this system is calculated, which is a Fomenko–Tsishang molecule, by the new method developed by the author.