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Let us write the equation as: $$x^3-a^{2}x+ax-x+a^2-a=0.$$ 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 $\boxed{0}$.