On the singular points of the orbifolds arising from integral, ternary quadratic forms

Mennicke (Proc R Soc Edinb Sect A 67:309–352, 1963; Proc R Soc Edinb Sect A 88:151–157, 1981) investigated the number of conic points with isotropies of orders 3,  4 and 6 that can appear in the orientable twofold orbifold covering of the orbifold Qf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{f}$$\end{document} associated with an integral ternary quadratic form. Mennicke made essential use of a theorem of Jones (The Arithmetic Theory of Quadratic Forms. The Carus Mathematical Monographs, vol. 10. MAA, Baltimore, 1950, Theorem 86). In this paper we revisit Mennicke’s results and we extend them to Qf,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{f},$$\end{document} that is, even when Qf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{f}$$\end{document} is non-orientable. Our method is new and independent of Jones’ Theorem. We also study the possible cusp points.