19.66 (2018)

We say that a variety $\Theta$ is of Tarski type if any two non-abelian $\Theta$-free groups of finite rank are elementarily equivalent.
$\qquad$ a) Find examples of Tarski type varieties distinct from the variety of all groups.
$\qquad$ b) Is it true that the Burnside variety $\mathfrak{B}_n$ of all groups of exponent $n$, where $n$ is big enough, is of Tarski type?
$\qquad$ c) Is it true that the $n$-Engel variety $\mathfrak{E}_n$ of all groups satisfying the identity $[[[x, y], y], \dots, y] \equiv 1$ ($n$ copies of $y$), where $n$ is big enough, is of Tarski type?