Strongly 2-perfect trail systems and related quasigroups

In a recent paper of the authors the question of for which values of m the quasigroups arising from strongly two 2-perfect m-cycle systems are the finite members of a variety was considered. In comparison with existing results on similar problems, it seems likely that for m prime, we shall fail to get varieties only for those values of m where strongly 2perfect closed m-trail systems which are not strongly 2-perfect m-cycle systems exist and we give a necessary and sufficient condition for their existence. We show that if there exist strongly 2-perfect closed m-trail systems satisfying an additional condition, then the quasigroups arising from strongly 2-perfect m-cycle systems are not the finite members of a variety. We give reasons why the existence of such systems seems plausible. We also show that, the quasigroups corresponding to strongly 2-perfect 9-cycle systems are not the finite members of a variety.