  A hamiltonian cycle system (HCS, for short) of a graph $\Gamma$ is a
partition of the edges of $\Gamma$ into hamiltonian cycles. A HCS is cyclic
when it is invariant under a cyclic permutation of all the vertices of
$\Gamma$; the existence problem for a cyclic HCS has been completely solved by
Buratti and Del Fra in 2004 when $\Gamma$ is the complete graph $K_v$, $v$ odd,
and by Jordon and Morris in 2008 when $\Gamma$ is the complete graph minus a
$1$-factor $K_v-I$, $v$ even. In this work we present a complete solution to
the existence problem of a cyclic HCS for $\Gamma = K_{m\times n}$, the
complete multipartite graph, when the number of parts $m$ is even. We also give
necessary and sufficient conditions for the existence of a cyclic and symmetric
HCS of $\Gamma$; the notion of a symmetric HCS of a graph $\Gamma$ has been
introduced in 2004 by Akiyama, Kobayashi, and Nakamura for $\Gamma =K_v$, $v$
odd, in 2011 by Brualdi and Schroeder when $\Gamma = K_v-I$, $v$ even, and,
very recently, by Schroeder when $\Gamma$ is the complete multipartite graph.
