  Assume that the local universal deformation of a Calabi-Yau 3-manifold X has
an automorphism which does not act by 1 or -1 on the third cohomology. We show
that the $F^2$ bundle in the Variation of Hodge structures of each maximal
family containing $X$ is constant in this case. Thus X cannot be a fiber of a
maximal family with maximally unipotent monodromy, if such an automorphism
exists. Moreover we classify the possible actions of such an automorphism on
the third cohomology, construct examples and show that the period domain is a
complex ball containing a dense set of complex multiplication points in this
case.
