Does $ \frac{z}{x+y}+\frac{y}{x+z} +\frac{z}{x-y}+\frac{y}{x-z}=9$ have a nonzero integer solution?

I came up with the following Diophantine equation:

$$ \frac{z}{x+y}+\frac{y}{x+z} +\frac{z}{x-y}+\frac{y}{x-z}=9, $$

where $x,y,z\in\mathbb Z,$ $xyz\ne0,$ and $x+y\ne0,$ $x-y\ne0,$ $x+z\ne0,$ $x-z\ne0.$

Does this equation have a nonzero integer solution?

If a solution exists, I would be interested in finding one, preferably a primitive solution satisfying $\gcd(x,y,z)=1$.

If no such solution exists, is there a proof of nonexistence?

I came up with this equation and became curious whether it has any nonzero integer solutions. I am particularly interested in whether the structure of the denominators (x\pm y) and (x\pm z) can be used to prove the existence or nonexistence of integer solutions.

I have not yet found a solution or a proof of nonexistence.

Any ideas or approaches would be appreciated.