  It is conjectured that for every pair $(\ell,m)$ of odd integers greater than
2 with $m \equiv 1\; \pmod{\ell}$, there exists a cyclic two-factorization of
$K_{\ell m}$ having exactly $(m-1)/2$ factors of type $\ell^m$ and all the
others of type $m^{\ell}$. The authors prove the conjecture in the affirmative
when $\ell \equiv 1\; \pmod{4}$ and $m \geq \ell^2 -\ell + 1$.
