Infinite $p$-groups containing exactly $p^2$ solutions of the equation $x^p=1$

We study arbitrary infinite 2-groups with three involutions and infinite locally finite $p$-groups ($p\ne2$), containing $p^2-1$ elements of order $p$. For odd $p$ the group $G=A\langle b\rangle$, where $A$ is a direct product of two quasicyclic 3-groups $|b|=9$, $b^3\in A$, and subgroup $A$ is generated by the elements of the commutator ladder of element $b$, is a unique infinite non-Abelian locally finite $p$-group whose equation $x^p=1$ has $p^2$ solutions.