The answer: Swanstone
How I found it: I got lucky. I just grabbed it and started taking it to local big-box places (I assumed it was a big-box as the materials in this house aren't necessarily 'high end').
I stopped at Lowes and the initial guy in the kitchen section had no clue, but I was fortunate that a curious co-worker popped his head over the counter, thought for a minute, then declared it 'Bermuda Sand' from 'Swanstone'. We found the swatches and, sure enough, he was right-on. So kudos tot he very helpful Lowes employee.
Apparently, these type of shower systems are just large panels of solid surface material that is glued up on the wall. They idea is that the panels abut, then caulked. The corner cove is more of an 'extra' piece that can be used if desired. The underlayment SHOULD be your standard tile surface...water proof cement board that is then sealed with some sort of water sealant. However, I don't know that in this case, so this particular piece seems to be a rather critical for my shower. So, glad we were able to find it.
The challenge for future people reading this is that I guess the solid surface companies are many, and the patterns all tend to be unique to each company, so it is a bit of a needle-in-a-haystack hunt trying to find these.
For contractors, I'd like to suggest leaving a note on the inside of a bath or kitchen vanity with the manufacturer's info. That'd be handy. ;)
I found the problem. Debris from a ruptured water filter (for the entire house) had come through the pipe and partially blocked a small filter between the pipe and the plastic T that holds the pivot ball. When I completely remove the shower head (so that I just had a pipe coming out of the wall in the shower), I could see a small screen filter in the plastic T of the shower head, and it had some junk partially blocking it. When I remove that and re-attached the shower head, the problem went away.
Best Answer
You were not very clear with your question.
I am assuming that you are looking for the fitting to which the sprayer head attaches, which then drops over the bracket peg in your first picture.
It is called a swivel connector. They are not "generic", they are specific to the manufacturer of your hand-held shower equipment. This one is from Alsons, a very common brand in the U.S.A. and the one I think you may have. Do an internet search for Alsons Swivel Connector.