$x^6 y + x^6 - x^5 y^2 + x^4 y^2 + x^4 y - x^3 y^4 - x^3 y^3 + x^3 y^2 + x^2 y^5 + x^2 y^3 - x y^5 = z, x > y$
... over the natural numbers?
Just curious what you call this kind of polynomial equation, as I am a non-mathematician doing some learning and research outside of any kind of school.
This is just the expansion of $$-a \left(a^2-a b+b\right) \left(b^2-a b-a\right) \left(a^2+a b+b^2\right)$$