The salamanders themselves don’t do photosynthesis, but they host symbiotic algae that do.
Spotted salamanders, too, are in a long-term relationship with photosynthetic algae. In 1888, biologist Henry Orr reported that their eggs often contain single-celled green algae called Oophila amblystomatis. The salamanders lay the eggs in pools of water, and the algae colonise them within hours.
By the 1940s, biologists strongly suspected it was a symbiotic relationship, beneficial to both the salamander embryos and the algae. The embryos release waste material, which the algae feed on. In turn the algae photosynthesise and release oxygen, which the embryos take in. Embryos that have more algae are more likely to survive and develop faster than embryos with few or none.
Then in 2011 the story gained an additional twist. A close examination of the eggs revealed that some of the algae were living within the embryos themselves, and in some cases were actually inside embryonic cells. That suggested the embryos weren’t just taking oxygen from the algae: they might be taking glucose too. In other words, the algae were acting as internal power stations, generating fuel for the salamanders.
–Marshall (2013): Zoologger: The first solar-powered vertebrate in New Scientist based on Graham et al. (2012).