Algebras with parastrophically uncancellable quasigroup equations

We consider $48$ parastrophically uncancellable quadratic functional equations with four object variables and two quasigroup operations in two classes: balanced non-Belousov (consists of $16$ equations) and non-balanced non-gemini (consists of $32$ equations). A linear representation of a group (Abelian group) for a pair of quasigroup operations satisfying one of these parastrophically uncancellable quadratic equations is obtained. As a consequence of these results, a linear representation for every operation of a binary algebra satisfying one of these hyperidentities is obtained.