Existence of 1-rotational k-cycle systems of the complete graph

We explicitly solve the existence problem for 1-rotational k-cycle systems of the complete graph $K_v$ with $v \equiv 1$ or k (mod 2k). For $v \equiv 1$ (mod 2k) we have existence if and only if k is an odd composite number. For any odd k and $v \equiv k$ (mod 2k), (except k = 3 and $v \equiv 15, 21$ (mod 24)) a 1-rotational k-cycle system of $K_v$ exists.