$xy(x + y) = yz(y + z) = zx(z + x) \Longleftrightarrow \frac{x+y}{z} = \frac{y+z}{x} = \frac{z+x}{y} \Longleftrightarrow \frac{x+y+z}{z} = \frac{x+y+z}{x} = \frac{x+y+z}{y}$ So all triples $(x, y, z)$ that satisfy are $x + y + z = 0$ or $x = y = z$.