  The generalization of the Oberwolfach Problem, proposed by J. Liu in 2000,
asks for a uniform $2$-factorization of the complete multipartite graph
$K_{m\times n}$. Here we focus our attention on $2$-factorizations regular
under the cyclic group $Z_{mn}$, whose $2$-factors are disjoint union of cycles
all of even length $\ell$. In particular, we present a complete solution for
the extremal cases $\ell=4$ and $\ell=mn$.
