  In their paper Livn\'e and Yui (math.AG/0304497) discuss several examples of
non-rigid Calabi-Yau varieties which admit semi-stable K3-fibrations with 6
singular fibres over a base which is a rational modular curve. They also
establish the modularity of the L-function of these examples. The purpose of
this note is to point out that the examples which were listed in their paper,
but which do not lead to semi-stable fibrations, are still modular in the sense
that their L-function is associated to modular forms. We treat the case
associated to the group Gamma_1(7) in detail, but our technique also applies to
many other cases. We further make some comments concerning the Kummer
construction for fibre products of elliptic surfaces in general.
