Quite an Absolute question indeed.

Among all the solutions of this problem it is obvious that our two variables can only take values from $[-1;1]$ .This happens because into our equation our two factors in the left side are always going to be positive.Thus ,if the variables exceed the limitation above the right side of the equation is always going to be bigger than 1.Now ,after saying that we can continue solving the problem.According to our limits above we know that $x$ can only take these three values which are $-1$ , $0$ and $1$ and so for $y$ .Combining these values for our variables we can know calculate all the possible pairs as follows: $(0;1)$ , $(0;-1)$ , $(1;0)$ , $(-1;0)$ , $(1;1)$ and $(-1;-1)$ .Puting these pairs into our equation we see that all of them work perfectly,therefore our equation has $\boxed{6}$ solutions.