  Let M be a smooth connected compact surface, P be either the real line R^1 or
the circle S^1. For a subset X of M denote by D(M,X) the group of
diffeomorphisms of M fixed on X. In this note we consider a special class F of
smooth maps f:M\to P with isolated singularities which includes all Morse maps.
For each such map f from F we consider certain submanifolds X of M that are
"adopted" with f in a natural sense, and study the right action of the group
D(M,X) on C^{\infty}(M,P). The main result describes the homotopy types of the
connected components of the stabilizers S(f) and orbits O(f) for all maps f
from F. It extends previous author results on this topic.
