" 44 reported evidence for REM sleep in the chameleon lizard. They reasoned that this animal, having very mobile active eyes in waking would be more likely to show eye movement periods during sleep. They, like most subsequent researchers working with reptiles, found relatively little modulation of forebrain EEG across the sleep-wake cycle, compared to the dramatic modulation in mammals. Some spiking occurred in forebrain leads during sleep, but no change in EEG occurred during periods of rapid eye movement. In "REM" sleep, one eye could be open while the other remained shut. No change in muscle tone occurred during sleep. No arousal threshold testing is described in this brief report. " ...
https://www.semel.ucla.edu/sites/default/files/sleep/pdf/rem-evolution.pdf
"Some REM characteristics have been found in reptiles, including chameleons, desert iguanas and caimans. But the experiments all had problems that have left the question open."...
https://www.dovepress.com/the-influence-of-gravity-on-rem-sleep-peer-reviewed-fulltext-article-OAAP
"they demonstrated that typically brief periods of REM are present while sleeping. Unilateral eye movements with the eyelid opened are also observed without any changes in sleep posture. However, their electroen- cephalographic recordings from the telencephalon suggested no true REM sleep is present. Given the radically di erent anatomy of the lizard brain, recordings from the brain stem (where the structures homologous to those recorded in mammals and birds reside) would be needed to con rm this. In conclusion, neurophysiological studies in reptiles are woefully lacking behind those of other vertebrates and much remains to be investigated." ...
http://www.anthonyherrel.fr/publications/Herrel 2013 Physiology - Chameleon Book.pdf