$(0, 0, 0)$ is an obvious solution. We are going to show that it is the only solution. This can be shown by using the infinite descending method.

Assume $(x, y, z)$ is a not-all-zero solution to the given equation which minimize the quantity of $x+y+z$. Clearly $x$ is an even number, let it be $x = 2x_1$. Then $x_1 < x$. Replacing $x$ with $x_1$ and simplifying in the above equation gives $$4x_1^3 + y^3 = 2z^3$$

Therefore, $y$ must be even. Let $y=2y_1$ where $y_1 < y$. It follows that $$2x_1^3+4y_1^3=z^3$$

Hence, $z$ must be even as well. Let $z=2z_1$ where $z_1 < z$. Now, we conclude if $(x, y, z)$ is a solution, so will be $(x_1, y_1, z_1)$. However, the fact of $x_1 + x_2+x_3 < x+ y +z$ contradicts the assumption that $x+y+z$ is the smallest among all non-zero solutions.