On the exponential Diophantine equation $(a(a-l)m^2+1{)}^x+(al{m}^2-1{)}^y=(am{)}^z$

Suppose that $a$, $l$, $m$ are positive integers with $a\equiv 1(\mathrm{mod}2)$ and $a^2{m}^2\equiv -2(\mathrm{mod}p)$, where $p$ is a prime factor of $l$. In this paper, we prove that the title exponential Diophantine equation has only the positive integer solution $(x,y,z)=(1,1,2)$. As an another result, we show that if $a=l$, then the title equation has positive integer solutions $(x,y,z)=(n,1,2)$, $n\in \mathbb{N}$. The proof is based on elementary methods, Bilu-Hanrot-Voutier Theorem on primitive divisors of Lehmer numbers, and some results on generalized Ramanujan-Nagell equations.