On $\pi$-quasigroups of type $T_1$

Quasigroups satisfying the identity $x(x\cdot xy)=y$ are called $\pi$-quasigroups of type $T_1$. The spectrum of the defining identity is precisely $q=0$ or $1\pmod3$, except for $q=6$. Necessary conditions when a finite $\pi$-quasigroup of type $T_1$ has the order $q=0\pmod3$, are given. In particular, it is proved that a finite $\pi$-quasigroup of type $T_1$ such that the order of its inner mapping group is not divisible by three has a left unit. Necessary and sufficient conditions when the identity $x(x\cdot xy)=y$ is invariant under the isotopy of quasigroups (loops) are found.