Varieties of quasigroups arising from 2-perfect m-cycle systems

For m = 6 and for all odd composite integers m, as well as for all even integers m ≥ 10 that satisfy certain conditions, 2-perfect m-cycle systems are constructed whose quasigroups have a homomorphism onto quasigroups which do not correspond to a 2-perfect m-cycle systems. Thus it is shown that for these values of m the class of quasigroups arising from all 2-perfect m-cycle systems does not form a variety.