Existence of cyclic k-cycle systems of the complete graph

Starting from earliest papers by Rosa we solve, directly and explicitly, the existence problem for cyclic k-cycle systems of the complete graph $K_v$ with $v \equiv 1$ (mod 2k ), and the existence problem for cyclic k-cycle systems of the complete m-partite graph $K_{m×k}$ with m and k being odd. As a particular consequence, a cyclic p-cycle system of $K_v$ with p being a prime exists for all admissible values of v but $(p,v)̸\neq(3,9)$.This was previously known only for p=3, 5, 7.