What does the constraint on the signed support vectors in SVMs signify?

What does the constraint $\sum_i \alpha_i y_i = 0$ on the support vectors signify? Does it mean a data set cannot have only one support vector? Can all the support vectors of a data set after classification belong to only one class (because, then the above constraint seems to be violated as $\alpha_i \geq 0$)? If not, how can we say, we can always find two support points belonging to different classes - with the same distance from the learned SVM, requiring maximum marginalization? Although this seems to be depending on the data set, the constraint needs to be true irrespective of the form of the data.