On subgroups of free Burnside groups of odd exponent $n\geqslant 1003$

We prove that for any odd number $n\geqslant 1003$, every non-cyclic subgroup of the 2-generator free Burnside group of exponent $n$ contains a subgroup isomorphic to the free Burnside group of exponent $n$ and infinite rank. Various families of relatively free $n$-periodic subgroups are constructed in free periodic groups of odd exponent $n\ge 665$. For the same groups, we describe a monomorphism $\tau$ such that a word $A$ is an elementary period of rank $\alpha$ if and only if its image $\tau(A)$ is an elementary period of rank $\alpha+1$.