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