In  we studied equivariant bifurcation
problems with a symmetry group acting
on parameters, from the point of view of singularity theory. We followed
classical theory originated by Damon ,
using the ideas presented in [5, 13, 14].
We adapted general results about unfoldings, the algebraic characterization
determinacy, and the recognition problems, to multiparameter bifurcation
f(x, λ)=0 with ‘diagonal’
symmetry on both the state variables and on the
bifurcation parameters. More precisely, such bifurcation problems satisfy
for all γ∈Γ, where Γ is a compact Lie group
In this paper we attack the same problem from a different angle: the
path formulation. This idea can be traced back to the first papers
of Mather  and Martinet
[15, 16]. It was used explicitly in
Golubitsky and Schaeffer  (see also their earlier
paper ) as a way of relating bifurcation
problems in one state variable without
symmetry to a miniversal unfolding in the sense of catastrophe theory.
At that time
the techniques of singularity theory were not powerful enough to handle
power of the idea efficiently – either in theory or in computational
is why the path formulation was abandoned in favour of contact equivalence
distinguished parameters, as developed in Golubitsky and
Schaeffer . Considerable
progress has been made since then; for example Montaldi and Mond
the path formulation to apply the idea of [Kscr ]V-equivalence
introduced by Damon 
to equivariant bifurcation theory. Bridges and Furter 
studied equivariant gradient
bifurcation problems using the path formulation, and defined an equivalence
relation in the space of paths and their unfoldings that respects contact
of the gradients. Here we describe an algebraic approach to the path formulation
that has the advantage of organizing the classification of normal forms.
it minimizes the calculation involved in obtaining the normal forms (compare
the classical framework in Furter et al. ).
The geometric approach to the path
formulation using [Kscr ]V-equivalence is still open
in the context of a symmetry group acting diagonally on parameters.