Analogues of Nielsen’s and Magnus’s theorems for free Burnside groups of period $3$

We prove that the free Burnside groups $B(m,3)$ of period 3 and rank $m\geq1$ have Magnus's property, that is if in $B(m,3)$ the normal closures of $r$ and $s$ coincide, then $r$ is conjugate to $s$ or $s^{-1}$. We also prove that any automorphism of $B(m,3)$ induced by a Nielsen automorphism of the free group $F_m$ of rank $m$. We show that the kernel of the natural homomorphism $\mathrm{Aut}(B(2,3)) \rightarrow GL_2(\mathbb{Z}_3)$ is the group of inner automorphisms of $B(2,3)$.