  Due to a result by Glasner and Downarowicz, it is known that a minimal system
is mean equicontinuous if and only if it is an isomorphic extension of its
maximal equicontinuous factor. The majority of known examples of this type are
almost automorphic, that is, the factor map to the maximal equicontinuous
factor is almost one-to-one. The only cases of isomorphic extensions which are
not almost automorphic are again due to Glasner and Downarowicz, who in the
same article provide a construction of such systems in a rather general
topological setting.
  Here, we use the Anosov-Katok method in order to provide an alternative route
to such examples and to show that these may be realised as smooth skew product
diffeomorphisms of the two-torus with an irrational rotation on the base.
Moreover - and more importantly - a modification of the construction allows to
ensure that lifts of these diffeomorphism to finite covering spaces provide
novel examples of finite-to-one topomorphic extensions of irrational rotations.
These are still strictly ergodic and share the same dynamical eigenvalues as
the original system, but show an additional singular continuous component of
the dynamical spectrum.
