Let us write the equation as:
$$x^3 - a^2x + ax - x + a^2 - a = 0.$$
Group the $x$ terms:
$$x^3 + (a - a^2 - 1)x + (a^2 - a) = 0.$$
This is a cubic equation of the form:
$$x^3 + px + q = 0,$$
where $p = a - a^2 - 1$ and $q = a^2 - a$.

By Vieta's formulas, the sum of the roots of the cubic equation $x^3 + px + q = 0$ is $0$ (since the coefficient of $x^2$ is $0$).

Therefore, the sum of all three solutions is $\boxed{0}$.