Ah, how the road to mathematical truth is paved in blood...
There are more solutions than you should care to count. A simple calculator program can tell you that...
etcetera ad nauseam
As you know, Diophantine equations are notoriously difficult to solve. There may be an infinite number of solutions readily found by number-crunching.
What exactly is it that you want to know about this equation? A proof of whether or not there is an infinitude of points? A formula generating a set of solutions?
Here's a small part:
If you rearrange the equation you'll get the following:
When X=1, and Y=-1 we get , in other words
So, any Z will work for X=1 and Y=-1
Some other brute force stuff:
When you set one variable to equal zero, there's only a finite number of solutions that will work (tried for X = 0 and Z = 0, Y = 0 will probably behave the same way)
I don't think there's one general formula for the solution though, because of the (1,-1,Z) case ...