The continuation of symmetric periodic orbits can be described in terms of “symmetry functions”; the branching of the zero-level lines in a neighbourhood of a critical point gives rise to the transition from “first kind” to “second kind” periodic orbits. When the families are parametrized with the Jacobi integral, the bifurcations can be described as “catastrophes” of the generating functions. However bifurcations of higher order are more complex than the generic catastrophes with one parameter: both symmetric and asymmetric bifurcations occur.

In this way the symmetric periodic orbits that do not have close approaches to the secondary body can be described in an analytic way and their stability can be deduced from simple bifurcation rules. However numerical experiments are required to determine the “natural termination” of the families.