There is a whole youtube video by ironworker gaming that goes over the exotic.
Basically guns and even your abilities that apply burn don't make the burn last longer. Two tailed, prospector, polaris lance, and even your solar grenade and bottom tree melee dont even burn for longer. Mods like buring cells and environment damage don't even work ☹️.
And some of the guns that do apply burn damage basically don't even get your melee back faster. Like buring cells, prospector, prometheus, and joutunn.
This exotic is very buggy in general and don't work properly with the already niche things it is asking for.
Here is the video
https://youtu.be/pifJq-ZHVlc
-
I don't think it is supposed to work with solar burn weapons like burning edge on blade barrage