The paper deals with the topological classification of singularities of vector fields on the plane which are invariant under reflection with respect to a line. As it has been proved in previous papers, such a classification is necessary to determine the different topological types of singularities of vector fiels on R3 whose linear part is invariant under rotations. To get the classification we use normal form theory and the the blowing-up method.

We implement a singularity theory approach, the path formulation, to classify ${𝔻}_3$-equivariant bifurcation problems of corank 2, with one or two distinguished parameters, and their perturbations. The bifurcation diagrams are identified with sections over paths in the parameter space of a ${𝔻}_3$-miniversal unfolding $F_0$ of their cores. Equivalence between paths is given by diffeomorphisms liftable over the projection from the zero-set of $F_0$ onto its unfolding parameter space. We apply our results to degenerate...

$\mu$-constant families of holomorphic function germs with isolated singularities are considered from a global perspective. First, a monodromy group from all families which contain a fixed singularity is studied. It consists of automorphisms of the Milnor lattice which respect not only the intersection form, but also the Seifert form and the monodromy. We conjecture that it contains all such automorphisms, modulo $\pm id$. Second, marked singularities are defined and global moduli spaces for right equivalence...