  Given an irrational rotation $T$ on $\M T$ we settle necessary and sufficient
conditions on a step function $\phi$ and $t\in \M T$ for the existence of
measurable solutions to the cohomogical equation $$\exp{(2i\pi\phi)}=\e{2i\pi
t}f/f\rond T.$$ This yields a characterization of eigenvalues and
eigenfunctions for several transformations arising from irrational rotations
and step functions: cylinder flows, special flows, induced maps... From there
we give constructions of special flows and three-interval exchange
transformations with unusual spectral properties. In both cases we exhibit
examples with Kronecker factors of infinite rank. We also construct
three-interval exchange transformations which are non-trivially conjugate to
irrational rotations or to odometers. Similarly there exist special flows over
irrational rotations which are non-trivially conjugate to translations flows on
$\M T^2$ or on solenoids. Finally, we prove a regularization property which
allows us to give similar examples of special flows with smooth ceiling
functions, under natural Diophantine conditions for the rotation.
