Could you explain better what you are looking for ?
In fact, the set of all vectors v such that , for and eigenvalue of a A is a subspace. That's why you cannot solve for just a single pair of values for x and y. You arrived at y= x so <x, y>= <x, x>= x<1, 1> or all mutiples of <1, 1> are eigenvectors corresponding to eigenvalue 2. That is the "eigenspace" for eigenvalue 2.