Some bounds on the number of non isomorphic cyclic k-cycle systems of the complete graph

In a recent paper (2024) M. Buratti and M.E:Muzychuck have established some lower bounds on the number of non isomorphic cyclic Steiner Triple Systems of order $v\equiv 1$ (mod $6$). We complete their result to the case $v\equiv 3$ (mod $6$). For each odd $k>3$ we also find lower bounds for the number of non isomorphic cyclic $k$-cycle systems of a complete graph.