Odd vanishing cycles in cyclotomic fields

A cusp of a Hecke group $G_q$ has two natural lifts to the ring of integers of a cyclotomic field. These lifts are called here odd vanishing cycles. All lifts of all cusps together form a discrete subset of ${\mathbb C}$ of some exquisite beauty. They form one or two or four orbits of a certain subgroup of the matrix Hecke group. The subgroup can be considered as a monodromy group and is an analog of a rank 2 Coxeter group, so of a dihedral group. The paper has a research part and a larger survey part.