\(k\)-perfect \(3k\)-cycle systems

The spectrum for \(k\)-perfect \(3k\)-cycle systems is considered here for arbitrary \(k \not\equiv 0 \pmod{3}\). Previously, the spectrum when \(k = 2\) was dealt with by Lindner, Phelps, and Rodger, and that for \(k = 3\) by the current authors. Here, when \(k \equiv 1\) or \(5 \pmod{6}\) and \(6k + 1\) is prime, we show that the spectrum for \(k\)-perfect \(3k\)-cycle systems includes all positive integers congruent to \(1 \pmod{6k}\) (except possibly the isolated case \(12k + 1\)). We also complete the spectrum for \(k = 4\) and \(5\) (except possibly for one isolated case when \(k = 5\)), and deal with other specific small values of \(k\).