  We study the connection between conjugations of a special kind of dynamical
systems, called P-configurations, and solutions to homogeneous Cauchy type
functional equations. We find that any two regular P-configurations are
conjugate by a homeomorphism, but cannot be conjugate by a diffeomorphism. This
leads us to the following conclusion (resolving an open problem posed by
Paneah): there exist continuous nonlinear solutions to the functional equation:
f(t) = f((t+1)/2) + f((t-1)/2), t \in [-1,1] .
