Converting a 6-cycle system into a steiner triple system

Given a 6-cycle system, two ways of transforming it into a Steiner triple system have been previously considered: one can either inscribe into each 6-cycle two triangles, or one can squash each 6-cycle into two triangles. In this paper, we consider yet another way, which we call converting: delete two opposite edges of each 6-cycle and add two short diagonals to create two triangles. If, when doing this to every 6-cycle of a 6-cycle system, it results in a Steiner triple system, we call the latter a converted 6-cycle system. We prove that a converted 6-cycle system of order v exists if and only if v ≡ 1 or 9 (mod 12), v ≥ 13. We also prove an analogous result for maximum packings of 6-cycles.