And yet vertebrates already share one, biliverdin. It’s the stuff that makes some eggs blue-green, duck and goose feces green, and then some lizards and fish have it in large enough amounts to have green blood. It’s a waste product but is repurposed for visual color in a lot of invertebrates like insects…among vertebrates what I could find suggests it might in a very few fish, like bluish varieties of perch, though like you say certainly not often.
But that does mean it isn’t actually the case that vertebrates can’t synthesize green pigments. They do and instead there is something else that keeps them from using them that way.
I was going to answer “well, yes, it isa waste product, and toxic in larger quantities”, but then, 1) this isn’t as clear any longer as it was during the time I was learning biochemistry and metabolic pathways, and 2) having a waste product would not necessarily mean it can’t be used in fur pigmentation…
Well, it isn’t really stable when exposed to light as far as I remember. That might be a good reason for many green pigments…
Are you sure it isn’t light stable? The related porphyrin pigments usually aren’t…though the best known example of vertebrates with a legitimately green pigment, turacos, have a porphyrin pigment containing copper. But biliverdin or its protein complexes definitely work for some worms, the odd green spider, and a lot of insects, so it can’t be that it isn’t chemically suited for the job.