  Let M be a smooth connected compact surface, P be either the real line R^1 or
the circle S^1, and f:M-->P be a smooth mapping. In a previous series of papers
for the case when f is a Morse map the author calculated the homotopy types of
stabilizers and orbits of f with respect to the right action of the
diffeomorphisms group of M. The present paper extends those calculations to a
large class of maps M-->P with degenerate singularities satisfying certain set
of axioms.
