  In this paper we present an explicit construction for the fundamental
solution to the Dirac and Laplace operator on some non-orientable conformally
flat manifolds. We first treat a class of projective cylinders and tori where
we can study monogenic sections with values in different pin bundles. Then we
discuss the M\"obius strip, the Klein bottle and higher dimensional
generalizations of them. We present integral representation formulas together
with some elementary tools of harmonic analysis on these manifolds.
