On groups with a splitting automorphism of prime order

An automorphism $\varphi$ of a group $G$ is called a splitting automorphism of prime order $p$, if $\varphi=1$ and $x\cdot x^{\varphi}\cdot x^{\varphi^2}\cdot\dots\cdot x^{\varphi^{p-1}}=1$ for all $x\in G$. In [E. I. Khukhro, “Locally nilpotent groups admitting a splitting automorphism of prime orded,” Mat. Sb., 130, No. 1, 120–127 (1986)] there was obtained a positive solution to the restricted Burnside problem for the variety $\mathfrak{M}_p$ of all groups with splitting automorphism of prime order p, by establishing that the locally nilpotent groups in $\mathfrak{M}_p$ form a subvariety $LN\mathfrak{M}_p$. We conjecture that $LN\mathfrak{M}_p$ is a join of the subvariety $\mathfrak{B}_p\cap LN\mathfrak{M}_p$ of gruops of prime exponent and the subveriety $\mathfrak{N}_{c(p)}\cap\mathfrak{M}_p$ of nilpotent groups of some $p$-bounded class. In the article the following result is proved in this direction: there exist $p$-bounded numbers $k(p)$ and $l(p)$ such that every group $G$ in $LN\mathfrak{M}_p$ satisfies the identities $\bigl[x_1^{p^{k(p)}},x_2^{p^{k(p)}},\dots,x_{h+1}^{p^{k(p)}}\bigr]=1$ which means that the subgroup $G^{p^{k(p)}}$ is nilpotent of class $h(p)$; i.e., $\gamma_{h(p)+1}\bigl(G^{p^{k(p)}}\bigr)=1$) и $[x_1,x_2,\dots,x_{h+1}]^{p^{l(p)}}=1$, where $h(p)$ is the Higman function the nilpotency class of a nilpotent group with regular automorphism of prime orded $p$.