Using the theory of linear forms in logarithms and a result due to Adachi, we show that the title equation has only the trivial positive integral solution under some conditions.
"On the exponential Diophantine equation $a^x + l^y = c^z$." Proc. Japan Acad. Ser. A Math. Sci. 77 (9) 151 - 154, Nov. 2001. https://doi.org/10.3792/pjaa.77.151