Need help with a faucet

[sunset on the dock]

Here is my bathroom faucet. I am not sure of the brand and if I could figure that out I might be able to research my problem. As you can see the handle is loose and rocks back and forth. I expected to see a small allen head screw on the faucet that I would loosen and then reseat the faucet. But there isn't. If I push down on the handle it does not seat securely and the problem remains. Any ideas would be appreciated. Thank you.