Almost Resolvable Directed \(m\)-cycle systems: \(m \equiv 3 (mod 6)\)

Let \(m \equiv 3 \pmod{6}\). We show that there exists an almost resolvable directed \(m\)-cycle system of \(D_n\) if and only if \(n \equiv 1 \pmod{m}\), except possibly if \(n \in \{3m+1, 6m+1\}\).