Necessary conditions of applicability of Gaussian elimination to systems of equations over quasigroups

Previously, in the process of investigating systems of equations over the given family ${\mathfrak{S}}\,$ of quasigroup operations, the author proved the following fact: applicability of Gaussian elimination to the systems considered requires that generalized distributivity and transitivity identities hold for the operations from ${\mathfrak{S}}$. The present paper describes all sets of operations that satisfy these identities. The result obtained allows one to conclude that Gaussian elimination is applicable only if the system of equations is linear or may be reduced to a linear system.