On regularity of Sylow $p$-subgroups of the Chevalley group of types $F_4, E_6$ over the ring $\mathbb{Z}_{p^m}$

In this paper, we find necessary and sufficient conditions for the regularity of the Sylow $p$-subgroup $P$ of the Chevalley group of types $F_4$ or $E_6$ defined over the ring of integers modulo $p^m$ when $p$ is a prime different from $37,41,43,47$. For the listed values of $p,$ the group $P$ is regular if the exponent $m$ does not exceed $3$; for $m$ greater than $3$, the answer remains unknown.