Answer: What kind of animal can do that?

The short answer: the Order is Caudata. 

As I mentioned, this challenge wasn't that difficult--but it IS surprising to find a complex animal that has two closely related forms with features not normally associated with vertebrates.  I was certainly surprised to find that there are salamanders without lungs and salamanders that are photosynthetic during a part of their lives.  

From a search perspective, this wasn't that hard.  I started with:

[ lungless vertebrate ]

and quickly found the lungless salamander link on Wikipedia.  I was, I have to admit, slightly skeptical, so I did a second search by picking up the scientific name of one of the lungless salamanders (Batrachoseps) and doing a search for it:

[ Batrachoseps ]

and finding all kinds of authoritative resources (mostly herpetology departments at universities), all of which describe it as lungless.  

Kaldari Batrachoseps attenuatus / Wikipedia image.

These relatively slender salamanders absorb oxygen directly from the air through their skin and the tissue lining their mouth, a remarkable development that provides an interesting thought about how lungs might have evolved.  

They also have a "naso-labial groove" that is lined with glands to help the animal with "chemoreception"--that is, with its sense of smell.  

We normally think about the ability to smell and the ability to breathe as inextricably linked, but for these salamanders, they're two very different systems.  

Now that I had a lungless salamander, I did the next obvious search:

[ photosynthetic salamander ]

and found that the embryos of the spotted salamander (Ambystoma maculatum) incorporate an algae (Oophila amblystomatis).  The remarkable insight was to discover that they don't just live together, but that the embryo relies on the oxygen and carbohydrates that are generated by the algal cells.  The salamander embryos rely on the green algae cells to make it through their first few weeks of life. 

Spotted salamander embryos /Ted Levin/PhotoLibrary.com

This is particularly interesting since many people have suspected that mitochrondria were originally bacteria that somehow became incorporated into animal cells.  This looks very much like this same process is going on here.  You could imagine an evolutionary story for salamanders that has a truly photosynthetic salamander in the distant future.  

One reader (with the unusual username "Me") pointed out that there are also some lungless frogs found in Borneo and that some frogs have algae in their egg masses as well.  So it's possible that some frogs might ALSO come in both lungless and photosynthetic forms.  (But I wasn't able to find a study that confirmed the frog embryos were dependent on the oxygen produced by the algae... so we don't know if they're photosynthetic or not!)  

But it was a nice find, Me.  And of course, it's easy to find the Order name for frogs:  Anura.  

And for all the readers who asked about why the picture of the sun rising over the water was a clue... it was simply a reminder than most salamanders and frogs need to spend some part of their life cycle in water (or at least very damp conditions).  

Search on!